The magnitude of the torque depends on how far out on the wrench the force is applied and on how much of the force is perpendicular to the wrench at the point of application. The number we use to measure the torque's magnitude is the product of the length of the lever arm r and the scalar component of F perpendicular to r.
In the notation of Figure Recall that we defined u X v to be 0 when u and v are parallel. This is consistent with the torque interpretation as well. If the force F in Figure The bar rotates counterclockwise around P. In this example the torque vector is pointing out of the page toward you.
The number Iw II cos I is the parallelepiped's height. Since the dot product is commutative, we also have 1 The dot and cross may be interchaoged in a 1riple scalar product without altering its value. Find the area of the triangle detennined by the points P, Q, andR.
Find a unit vector perpendicular to plane PQR. Given nonzero vectors n, v, and lV, use dot product and cross product notatioo, as appropriate, to describe the followiog. P 2,-2,1 , Q 3,-1,2 , R 3,-I, I The vector projection of u onto v b. What can you conclude about the associativity of the cross product? Parallel and perpendicular vectoR l. Whichveetors,ifany, are a perpendicular'l b Parallel? Parallel and perpendicular vectOR I.
A vector orthogonal to u X v and u X w f. A vector oflength Iu I in the direction of v v u c. A vector orthogonal to u X v and w d. The volwue of the parallelepiped detennined by u, v, and w Which of the following are always true, and which are not always true?
Let u, v, and w be vectors. Which of the followiog make seose, and which do not? Cross products of three vectOR Show that except in degenerate cases, u X v X w lies in the plane of u and v, whereas u X v X w lies in the plane ofv and w. What are the degenerate cases? Area of a Parallelogram Find the areas of the parallelograms whose vertices are given in Exercises Find the volume of a parallelepiped if four of its eight vertices are A O, 0, 0 , B l, 2, 0 , C O, -3,2 , and D 3, -4, 5.
Triangle area Find a concise formula for the area of a triangle in the xy-plane with vertices al. Triangle area Find a formula for the area of the triangle in the xy-plane with vertices at 0,0 , ai, az , and bl. Explain your work. Lines and Planes in Space We will use these representations throughout the rest ofthe book.
In space a line is detennined by a point and a veclor giving the direction of the line. Suppose that L is a line in space passing through a point Po xo, Yo, zo paralle.! The value of I depends on the location of the point P along the line, and the domain of I is - 00, In other words, the principal normal vector N will point toward the concave side of the curve Fignre Note: Since the curvature ofa curve remains the same if the curve is translated arrotatell, this result is true for any parabola.
As in Exercise 17, the same is true for any ellipse. What is the largest value" can have for a given value of b? We assome 8 to be a differentiable function of t for the motion. The law of conservation of energy tells us that the particle's speed after it has fallen straight down a distance z is v'2g. Express the particle's 8- and z-coordinates as functions of t. Does the aceeleration have any nonzero component in the direction of the binormal vector B?
With your computer or calculator in radian mode, use Newton's method to fmd the solution to as many places as you can. In Section Positive z-axis. Express the distance the particle travels along the helix as a function of8. Find the beetle's acceleration and velocity in polar form when it is halfway to 1 in.
To the nearest tenth of an inch, what will be the length of the path the beetie has traveled by the time it reaches the origin? Arc length in cylindrical coordinates a. Interpret this result geometrically in terms of the edges and a diagonal of a box. Sketch the box. Conservation of angular mom. Suppose the force acting on the object at time t is F I 9 , r, 9, 0 a.
Show tbat n,. In physics the angular momentum. Prove tbat angular momentum is a conaerved quantity; I. Parametric and Polar Equation. Moving in Three Dimensions Compute distance traveled, speed, curvature, and torsion for motion along a space curve. Visualize and compute the tangential, normal, and binormal vectors associated with motion along a space curve.
In this chapter we extend the basic ideas of single variable calculus to functions of several variables. Their derivatives are more varied and interesting because of the different ways the variables can interact. The applications of these derivatives are also more varied than for single-variable calculus, and in the next chapter we will see that the same is true for integrals involving several variables.
Real-valued functions of several independent real variables are defined similarly to functions in the single-variable case. Points in the domain are ordered pairs triples, quadruples, n-tuples of real numbers, and values in the range are real numbers as we have worked with all along. The set D is the function's domain. The set of w-values taken on by f is the function's range.
The symbol w is the dependent variable of f, and f is said to be a function of the n independent variables Xl to X n. We also call the Xj'S the function's input variables and call w the function's output variable. If f is a function oftwo independent variables, we usually call the independent variables X and y and the dependent variable z, and we picture the domain of f as a region in the xy-plane Figure If f is a function of three independent variables, we call the independent variables x, y, and z and the dependent variable w, and we picture the domain as a region in space.
In applications, we tend to use letters that remind us of what the variables stand for. In either case, rand h would be the independent variables and V the dependent variable of the function. Domains and Ranges In defining a function ofmore than one variable, we follow the usual practice of excluding inputs that lead to complex numbers or division by zero. The domain of a function is assumed to be the largest set for which the defining rule generates real numbers, unless the domain is otherwise specified explicitly.
The range consists of the set of output values for the dependent variable. Note the restrictions that may apply to their domains in order to obtain a real value for the dependent variable z.
Closed intervals [a, b] include their boundary points, open intervals a, b don't include their boundary points, and intervals such as [a, b are neither open nor closed. Functions of Several Variables A point xo, YO is a boundary point of R if every disk centered at xo, YO contains points tbat lie outside of R as well as points tbat lie in R.
The boundary point itself need not belong to R. The interior points of a region, as a set, make up the interior of the region. The region's boundary points make up its boundary. A region is open ifit consists entirely of interior points. A region is closed if it contains all its boundary points Figure An interior point is necessarily a point of R. The unit circle. Contains all boundary points. As with a half-open interval of real numbers [a, b , some regions in the plane are neither open nor closed.
If you start with the open disk in Figure The boundary points tbat are there keep the set from being open. The absence of the remaining boundary points keeps the set from being closed.
A region is unbounded if it is not bounded. Examples of unbounded sets in the plane include lines, coordinate axes, the graphs of functions defined on infinite intervals, quadrants, half-planes, and the plane itself. The points above the parabola make up the domain's interior. Graphs, Level Curves, and Contours of Functions of Two Variables There are two standard ways to picture the values of a function fix, y.
One is to draw and label curves in the domain on which f has a constant value. The set of all points x, y, f x, y in space, for x, y in the domain of f, is called the graph of f. It is still a level curve. Not everyone makes this distinction, however, and you may wish to call both kinds of curves by a single name and rely on context to convey which one you have in mind.
On most maps, for example, the curves that represent constant elevation height above sea level are called contours, not level curves Figure We can see how the function behaves, however, by looking at its threedimensional level surfaces. Washington in New Hampshire.
Reproduced by permission from the Appalachian Mountain Club. We are not graphing the function here; we are looking at level surfaces in the function's domain. The level surfaces show how the function's values change as we move through its domain. If we remain on a sphere of radius c centered at the origin, the function maintains a constant value, namely c.
If we move from a point on one sphere to a point on another, the function's value changes. It increases if we move away from the origin and decreases if we move toward the origin. The way the values change depends on the direction we take. The dependence of change on direction is important. We return to it in Section To accommodate the extra dimension, we use solid balls of positive radius instead of disks.
As with regions in the plane, a boundary point need not belong to the space region R. A point xo, Yo, zo is a boundary point of R if every solid ball centered at xo, Yo, zo contains points that lie outside of R as well as points that lie inside R Figure The interior of R is the set of interior points of R.
The boundary of R is the set of boundary points of R. A region is open if it consists entirely of interior points. A region is closed if it contains its entire boundary. A solid sphere Chapter PartiaL Derivatives with part of its boundary removed or a solid cube with a missing face, edge, or comer point is neither open nor closed.
Functions of more than three independent variables are also important. We can often get information more quickly from a graph than from a formula. At 25 ft, there is almost no variation during the year.
The graph also shows that the temperature 15 ft below the surface is about half a year out of phase with the surface temperature. When the temperature is lOM:st on the surface late January, say , it is at its highest 15 ft below; Fifteen feet below the ground, the seasons are reversed. Match each set of curves with the appropriate function. We refer to these level curves as a contour map. Does knowing that 21xyl- Irl;'. Show that liO, 0 and ly O, 0 exist, but I is not differentiable at 0,0.
The form depends on how many variables are involved, but once this is taken into account, it works like the Chain Rule in Section 3. The subscripts indicate where each of the derivatives is to be evaluated. Then add the products. With that understanding, we will use both of these forms interchangeably throughout the text whenever no confusion will arise.
The branch diagram in the margin provides a convenient way to remember the Chain Rule. The ''true'' independent variable in the composite function is t, whereas x and y are intermediate variables controlled by t and w is the dependent variable. Functions of Three Variables You can probably predict the Chain Rule for functions of three variables, as it only involves adding the expected third term to the two-variable formula. Read down eacb route, multiplying derivatives aloog the way; then add.
The branch diagram we use for remembering the new equation is similar as well, with three routes from w to t. Under the conditions stated below, w has partial derivatives with respect to both r and s that can be calculated in the following way.
The second can be derived in the same way, holding r fixed and treating s as t. The branch diagrams for both equations are shown in Figure The diagram for the second equation is similar; just replace r with s.
The preceding discussion gives the following. The branch diagram is shown in Figure Suppose that x , r aw ar aw a, dwax dx ar dwax dx aT dwax 1. The function F x, y is differentiable and 2. Computing the derivative ftmn the Chain Rule branch diagram in Figure Then at any point where Fy.. Give reasons for your x' - answer. What is the derivative of I in the direction of -i - 2j? Give reasons for yoar answer.
What is Vf at P? Du 1 1, -I is largest c. What is the derivative of I at P in the direction ofi n. Find the directions 0 D. Give reasons foryoor answer. I related to I.. In this direction, the value of the derivative is Theory and Examples Find the directions u and the values of D. Tangent Planes and Differentials In this section we derme the tangent plane at a point on a smooth surface in space.
Then we show how to calculate an equation of the tangent plane from the partial derivatives of the function defining the surface. This idea is similar to the dermition ofthe tangent line at a point on a curve in the coordinate plane for single-variable functions Section 3.
We then study the total differential and linearization of functions of several variables. The velocity vectors at Po therefore lie in a common plane, which we call the tangent plane at Po. Now let us restrict our attention to the curves that pass through Po Figure All the velocity vectors at Po are orthogonal to Vf at Po, so the curves' tangent lines all lie in the plane through Po normal to Vf.
We now define this plane. From Section The tangent plane is the plane through Po perpendicular to the gradient of f at Po. Fiod parametric equations for the lioe tangent to E at the poiot Po I, 1,3. The components ofv and the coordinates of Po give us equations for the lioe. In this case, we must fmd some other way to determine the behavior of I at a, b.
On the other hand, if the discriminant is negative at a, b , then the surface curves up in some directions and down in others, so we have a saddle point.
There are no local extreme values Example 2. Solution The function is defined and differentiable for all x and y and its domain has no boundary points. The function therefore has extreme values only at the points where Ix and I y are simultaneously zero.
Therefore, the point -2, -2 is the only point where I may take on an extreme value. The combination I", I I -2, -2 8. The two critical points are therefore 0, 0 and 2, 2. At the critical point 0, 0 we see that the value of the discriminant is the negative number , so the function has a saddle point at the origin. At the critical point 2, 2 we see that the discriminant has the positive value A graph of the surface is shown in Figure List the interior points ofR where I may have local maxima and minima and evaluate I at these points.
These are the critical points of I. List the boundary points ofR where I has local maxima and minima and evaluate I at these points.
We show how to do this shortly. Look through the lists for the maximum and minimum values of I. These will be the absolute maximum and minimum values of I on R. Since absolute maxima and minima are also local maxima and minima, the absolute maximum and minimum values of I appear somewhere in the lists made in Steps 1 and 2. SoLution Since I is differentiable, the only places where I can assume these values are points inside the triangle Figure The only interior point where! The maxilnum is 4, which I assumes at I, I.
The minimum is, which I assumes at 0, 9 and 9, 0. But sometimes we can solve such problems directly, as in the next example. EXAMPLE 6 A delivery company accepts only rectangular boxes the sum of whose length and girth perimeter of a cross-section does not exceed in. Find the dimensions of an acceptable box of largest volume. Let x, y, and z represent the length, width, and height ofthe rectangular box, respectively.
The volume is zero at 0, 0 , 0,54 , 54, 0 , which are not maximum values. Then Vyy V,. It does not apply to boundary points ofa function's domain, where it is possible for a function to have extreme values along with nonzero derivatives. Also, it does not apply to points where either I x or I y fails to exist. I x,y 8. I x,y 9. I x,y I x,y I Find the critical point of y finding Absolute ExtNma o :!
In Exercises , fmd the absolute maxima and minima of the :functions on the given doma. Find two munbers a and b with a :! Determine whether the function has a maximum, a minimlDn, or neither at the origin by imagining what the SUIface z - f x, y looks like. Describe your reasoning in each case. L f ;t,y - ;t1y2 b. Hint: Consider two cases: k - Oandk. A local minjmlDn at 0, O? For what values of k is the Second Derivative Test inconclusive? Give :reasons for your """"". Iff,, a,b - fy a,b - 0.
Can you. Find the point on the plane 3x origin. Find three numbers whose sum is 9 and whose sum ofsquares is a minimum. Find three positive numbers whose sum is 3 and whose product is a maximum. Among all closed rectangolar boxes of volume 27 em3, what is the smallest surface area?
You are to construct an open rectangular box from 12 It' of material. What dimensions will result in a box ofmaximum volume? Find the absolute maximum value of I over the square. As in any other single-variable case, the extreme values of I are then found among the values at the -I '" t '" 0 Functions: a. Find the minimum distance from the point 2, -I, I to the plane Find Many scientific calculators have these formulas built in, ensbling you to fmd m and b with only a few keystrokes after you have entered the data.
Finding a least squares line lets you 1. Use a CAS to perform the followiog steps: a. Plot the function over the given rectangle. Plot some level curves in the rectangle. Calculate the function's Hrst partial derivatives and use the CAS equation solver 10 rmd the critical points. How do the critical points relate 10 the level corves plotted in part b?
Which critical points, if any, appear to give a saddle point? Give reasons for d. Using the max-min tests, classifY the critical points found in part c. Are your rmdings consistent with your discussion in part c? If Ixxlyy - 1"12 4. The inequality we want comes from Equation 2. The last term is evaluated at a point on the line segment joining the origin aod x, y. Taylor's formula provides polynomial approximations of two-variable functions.
The f"lISt n derivative terms give the polynomial; the last term gives the approximation error. The first three terros of Taylor's formula give the function's linearization. To improve on the linearization, we add higher-power terms. How accurate is the approximation if Ixl ,,;; 0.
The error in the approximation is The third derivatives never exceed I in absolute value because they are products of sines aod cosines. Also, Ixl ,,;; 0. Hence IE x,y 1 ,,;; i«0. I 3 ,,;; 0. The error will not exceed 0. Estimate the error in the approximationiflxl'" 0. Use Taylor's formula to fmd a quadratic approximation of e r siny at the origin. Estimate the error in the approximation iflxl '" 0. In many applications, however, this is not the case.
In this section we learn how to find partial derivatives in situations like this, which occur in economics, engineering, and physics. Like many such systems, this one can be solved for two of the unknowns the dependent variables in terms of the others the independent variables. In being asked for iJwjax, we are told that w is to be a dependent variable and x an independent variable. The possible choices for the other variables corne down to Solution Dependent w,z w,y Independent x,y x,z In either case, we can express w explicitly in terms of the selected independent variables.
In the second case, where the independent variables are x and z and the remaining dependent variable is y, we eliminate the dependent variable y in the expression for w by replacing y' in the second equation by z This gives x'. The geometric interpretations of Equations I and 2 help to explain why the equations differ. As P moves on this parabola, w, which is the square of the distance from P to the origin, changes.
As P moves along this circle, its distance from the origin remains constant, and w, being the square of this distance, does not change. Decide which variables are to be dependent and which are to be independent.
In practice, the decision is based on the physical or theoretical context of our work. In the exercises at the end of this section, we say which variables are which. Eliminate the other dependent variable s in the expression for w. Differentiate as usual. Ifwe cannot carry out Step 2 after deciding which variables are dependent, we differentiate the equations as they are and try to solve for awjax afterward. The next example shows how this is done. It is not convenient to eliminate z in the expression for w.
We therefore differentiate both equations implicitly with respect to x, treating x and y as independent variables and w and z as dependent variables. Find 3. Show that Suppose that x 2 nates. What is a real-valued function of two independent variables? Three independent variables? What does it mean for sets in the plane or in space to be open? Give examples of sets that are neither open nor closed. How can you display the values ofa function f x,y of two independent variables graphically?
How do you do the same for a function f x,y, z of three independent variables? What are the basic properties of limits of functions of two independent variables? When is a function of two three independent variables continuous at a point in its domain? Give examples of functions that are continuous at some points but not others. What can be said about algebraic combinations and composites of continuous functions? Explain the two-path test for nonexistence of limits. How are they interpreted and calculated?
How does the relation between rITst partial derivatives and continuity of functions of two independent variables differ from the relation between rITst derivatives and continuity for real-valued functions of a single independent variable?
What is the Mixed Derivative Theorem for mixed second-order partial derivatives? How can it help in calculating partial derivatives ofsecond and higher orders? What does it mean for a function f x, y to be differentiable? What does the Increment Theorem say about differentiability? What is the relation between the differentiability off and the continuity of f at a point? What is the general Chain Rule?
What form does it take for functions of two independent variables? Functions dermed on surfaces? How do you diagram these differ- What is the derivative ofa function f x,y at a point Po in the direction of a unit vector n?
What rate does it describe? What geometric interpretation does it have? What is the gradient vector of a differentiable function f x, y? How is it related to the function's directional derivatives?
State the analogous results for functions of three independent variables. How do you rmd the tangent line at a point on a level curve of a differentiable function f x, y? How do you rmd the tangent plane and normal line at a point on a level surface of a differentiable function f x, y, z?
How can you use directional derivatives to estimate change? How do you linearize a function f x, y of two independent variables at a point xo, Yo? Why right you want to do this? How do you linearize a function of three independent variables?
What can you say about the accuracy of linear approximations of functions of two three independent variables? If we reverse the order of integration and attempt to calculate 1. Y we run into a problem because! There is no general role for predicting which order of integration will be the good one in circumstances like these.
If the order you rlISt choose doesn't work, try the other. Regions that are more complicated, and for which this procedure fails, can often be split up into pieces on which the procedure works. Sketch the region ofintegration and label the bounding curves Figure Find the y-limits ofintegration. Imagine a vertica1line L cutting through R in the direction of increasing y.
Mark the y-vaiues where L enters and leaves. These are the y-limits of integration and are usually functions of x instead of constants Figure 3. Find the x-limits of integration. The integral shown here see Figure The region of integration is given by the inequalities x 2 :5 y :5 2x and o :5 x :5 2. To find limits for integrating in the reverse order, we imagine a horizontal line passing from left to right through the region.
The integral is Solution Properties of Double Integrals Like single integrals, double integrals of continuous functions bave algebraic properties that are useful in computations and applications. Property 4 assumes that the region of integration R is decomposed into nonoverlapping regions R, and R2 with boundaries consisting of a finite number of line segments or smooth curves. Double Integrals over General Regions It follows that the constant multiple property carries over from sums to double integrals.
The other properties are also easy to verify for Riemann sums, and carryover to double integrals for the same reason. While this discussion gives the idea, an actual proof that these properties hold requires a more careful analysis of how Riemann sums converge. SoLution Figure So we choose to integrate in the order dx dy, which requires only one double integral whose limits of integration are indicated in Figure The regions D over which continuous functions are integmble are those having ''reasonably smooth" boundaries.
Volume of a Region in Space If Fis the constant function whose value is 1, then the sums in Equation 1 reduce to As dxk' d Yk, and dZk approach zero, the cells d Vk become smaller and more numerous and ml up more and more of D. D TIris detmition is in agreement with our previous detmitions ofvolume, although we omit the verification ofthis fact.
As we see in a moment, this integral enables us to calculate the volumes of solids enclosed by curved surfaces. Finding Limits of Integration in the Order dz dy dx We evaluate a triple integml by applying a three-dimensional version of Fubini's Theorem Section As with double integmls, there is a geometric procedure for finding the limits ofintegmtion for these single integmls.
To evaluate iff F x,y,z dV D over a region D, integrate first with respect to z, then with respect to y, and finally with respect to x. You might choose a different order ofintegmtion, but the procedure is similar, as we illustmte in Example 2. Sketch the region D along with its "shadow" R vertical projection in the xy-plane.
Label the upper and lower bounding surfaces of D and the upper and lower bounding curves of R. Triple Integrals in Rectangular Coordinates Find the z-limits ofintegration. Draw a line M passing through a typical point x, y in R parallel to the z-axis. These are the z-limits of integration. Draw a line L through x, y parallel to the y-axis.
These are the y-limits of integration. Find the x-limits ofintegration. These are the x-limits of integration. I x,y Follow similar procedures if you change the order of integration. The "shadow" of region D lies in the plane of the last two variables with respect to which the iterated integration takes place. The preceding procedure applies whenever a solid region D is bounded above and below by a surface, and when the "shadow" region R is bounded by a lower and upper curve.
It does not apply to regions with complicated holes through them, although sometimes such regions can be subdivided into simpler regions for which the procedure does apply. To find the limits of integration for evaluating the integral, we first sketch the region. The surfaces Figure Now we find the z-limits of integration.
J -- I I Finally we find the x-limits of integration. Use the order of integration dy dz dx. First we find the y-limits of integration. Next we find the z-limits of integration. SoLution First we find the z-limits of integration. Next we find the y-limits of integration. If F x,y, z is the temperature at x,y, z on a solid that occupies a region D in space, then the average value of F over D is the average temperature of the solid.
Solution We sketch the cube with enough detail to show the limits of integration Figure We then use Equation 2 to calculate the average value of F over the cube. In evaluating the integral, we chose the order dx dy dz, but any of the other five possible orders would have done as well.
Simply replace the double integrals in the four properties given in Section Evaluate one of the integrals. Evaluate the integrals in Exercises Let D be the region in Exercise Set up the triple integrals in spherical coordinates that give the volume of D using the following orders of integration. Then c find V. Let D be the smaller cap cut from a solid ball of radius 2 units by a plane 1 unit from the center of the sphere.
Express the volume of D as an iterated triple integral in a spherical, b cylindrical, and c rectangular coordinates. Then d find the volume by evaluating one of the three triple integrals. Finding TripLe IntegraLs Express the Moment of inertia of solid cone Find the moment of inertia of a right circular cone of base radius I and height I about an axis through the vertex parallel to the base.
Moment of inertia of solid sphere Find the moment of inertia ofa solid sphere of radius a about a diameter. Moment of inertia of solid cone Find the moment of inertia of a right circular cone of base radius a and height h about its axis. Hint: Place the cone with its vertex at the origin and its axis along the z-axis. Find the center of mass and the moment of inertia about the z-axis if the density is a.
Find the center of mass and the moment of inertia about the z-axis if the density is Variable density A solid ball is bounded by the sphere p Find the moment of inertia about the z-axis if the density is x' - x' - Find the center of mass.
Centroid x' Find the centroid of the solid in Exercise Thus, the centroid of a solid hemisphere lies on the axis of symmetry threeeighths of the way from the base to the top. Centroid of solid cone Show that the centroid of a solid right eircular eone is one-fourth of the way from the base to the vertex. Density of eenter of a planet A planet is in the shape of a sphere of radius R and total mass M with spherically symmetric density distribution that increases linearly as one approaches its center.
What is the density at the center of this planet if the density at its edge surface is tsken to be zero? Mass of planet's atmosphere A spherical planet of radius R bas an atmosphere whose density is p. In general, nonlinear transfunnations are more complex to analyze than linear ooes, and a complete treatment is left to a more advanced course. Notice as we move counterclockwise around the region R, we also move counterclockwise around the region G.
G The transfonned integrand function is easier to integrate than the original one, so we proceed to detennine the limits ofintegratioo for the transfonned integral.
As we move counterclockwise around the boundary of the region R, we also move counterclockwise around the boundary of G, as shown in Figure The method is like the method for double integrals except that now we work in three dimensions instead of two. As in the twodimensional case, the derivation of the change-of-variable formula in Equation 7 is omitted. For cylindrical coordinates, r, 8 , and z take the place of u, v, and w. We can drop the absolute value signs because sin cP is never negative for 0 ; cP ; 'IT.
Note that this is the same result we obtained in Section Here is an example of another substitution. Although we could evaluate the integral in this example directly, we have chosen it to illustrate the substitution method in a simple and fairly intuitive setting.
In this case, the bounding surfaces are planes. To apply Equation 7 , we need to find the corresponding uvw-region G and the Jacobian of the transformation. To find them, we first solve Equations 8 for x, y, and z in terms of u, v, and w.
Sketcb the transformed regioo in the uv-plane. Sketch the transformed region in the uv-plane. Sketch the 4. Sketch the transformed regioo in the uv-plane. Substitutions In Double Integrals 5. Vse the transfonnation in Exercise 3 to evaluate the integral Substitutions in Multiple Integrals Vse the transformation and parailelogramR in Exercise 4 to evaluate the integral Finding Jacobians a.
Then evaluate the uv-integral over G. Evaluating the integral directly requires a trigonometric substi1ution. Find the frrst moment of the plate about the origin. Evaluate the integral in Example 5 by integrating with respect to dydx into an integral over G, and evaluate bo1h integrals. Find the area this way.
Hint: Let x You can do this without evaluating any of the integrals. Cylindrieal sheIls In Section 6. Derme the double integral of a function of two variables over a bounded region in the coordinate plane. How are double integrals evaluated as iterated integrals? Does the order of integration matter? How are the limits of integration determined? How are double integrals used to calculate areas and average values. How can you change a double integral in rectangular coordinates into a double integral in polar coordinates?
Why might it be worthwhile to do so? Derme the triple integral of a function f x, y, z over a bounded region in space. How are double and triple integrals in rectangular coordinates used to calculate volumes, average values, masses, moments, and centers of mass?
How are triple integrals dermed in cylindrical and sphetical coordinates? Why might one prefer working in one of these coordinate systems to worldng in rectangular coordinates? How are triple integrals in cylindrical and sphetical coordinates evaluated?
How are the limits ofintegration found? How are substitutions in double integrals pictured as transformations of two-dimensional regions? Give a sample calculation. How are substitutions in triple integrals pictured as transformations of three-dimensional regions?
How are triple integrals in rectangular coordinates evaluated? Then evaluate both integrals. Average Values Evaluate the integrals in Exercises Area bonnded by lines and parabola Find the area of the ''tri- 2'If sm'lfX. First quadrant The triangle with vertices 0, 0 , 1, 0 , The first quadrant of the xy-plane.
Rectangular to cylindrical coordinates a Convert to cylindrical coordinates. Then b evaluate the new integral.
Then d find the integral of j by evaluating one of the triple integrals. Then c evaluate one of the integrals. Rectangular to spherical coordinates a Convert to spherical coordinates. In x -V! Average Practice Exercises 3 r sin 0 cos O z2 dz dr dO. Arrange the order of integration to be z first, then y, then x.
Rectangular to cylindrical coordinates The volume of a solid is 1 o 1T x 2 y Describe the solid by giving equations for the surfaces that form its boundary. Convert the integral to cylindrical coordinates but do not evaluate the integral. Spherical versus cylindrical coordinates Triple integrals involving spherical shapes do not always require spherical coordinates for convenient evaluation.
Some calculations may be accomplished more easily with cylindrical coordinates. Masses and Moments Moment orinertia of a "thick" sphere Find the moment of inertia of a solid of constant density 8 bounded by two concentric spheres ofradii a and b a As you will see, it does not matter where on the line this vertex lies. All such triangles have the same moment of inertia about the x-axis.
Then use the formula for Ix to fmd Iy and add the two to find Inertial moment Find the moment of inertia about the x-axis of a thin plate of constant density 8 covering the triangle with vertices 0, 0 , 3, 0 , and 3, 2 in thexy-plane.
Plate with variable density Find the center of mass and the moments of inertia about the coordinate axes of a thin plate Chapter b. Sketch the region and show the centroid in your sketch.
Substitutions What relationship must hold between the constants a, b, and c to mske Hint: l. Then c fmd the volume. Solid cylindrical region between two plane. Water in a hemispherical bowl A hemispherical bowl of radius 5 em is filled with water to within 3 em of the top.
Find the volume of water in the bowl. Transforming a double integral to obtain constant limits Sometimes a multiple integral with variable limits can be changed into one with constant limits. Hole in sphere A circular cylindrical hole is bored through a solid sphere, the axis of the hole being a diameter of the sphere.
Find the radius of the hole and the radius of the sphere. Evaluate the integral. What value of a will minimize the plate's polar moment of inertia about the origin? Mass and polar inertia of a counterweight The counterweight of a flywheel of constant density I has the form of the smaller segment cut from a circle of radius a by a chord at a distance b from the center b Changing the Order of Integration Evaluate the integral 1 00 e-ax - o dx.
Evaluate b e-XYdy a to form a double integral and evaluate the integral by changing the order of integration. Rewrite the Cartesian integral with the order of integration reversed. If you have not yet done Exercise 41 in Section Give reasons for steps i through iv. Integrate 0' over the plate to fmd the total charge Q. You plan to calibrate the bowl to make it into a rain gauge. What height in the bowl would correspond to 1 in. Its axis of symmetry is tilted 30 degrees from the vertical.
When it applies, Equation I can be a time-saver. Use it to evaluate the following integrals. Jo Jo hold. Hint: Put your coordinate system so that the satellite dish is in "standsrd position" and the plane of the water level is slanted Caution: The limits of integration are not "nice.
Set up, but do not evaluate, a triple integral in rectangular coordinates that gives the amount ofwaler the satellite dish will b. What would be the antallest tilt ofthe satellite dish so that it holds no water? An infinite balf-eylinder Let D be the interior of the infmite right circular half-cylinder ofradius I with its single-end face suspended 1 unit above the origin and its axis the ray from 0, 0, I to Use cylindrical coordinates to evaluate iff The gamma function, 1 ,..
Of particular interest in the theory of differential equations is the number Chapter J: dx Hypenolnme We have learned that 1 is the length of the interval [a, b] on the number line one-dimensional space ,! Mastering a beautiful subject with practical applications to the world is its own reward, but the real gift is the ability to think and generalize.
We intend this book to provide support and encouragement for both. We have made numerous revisions to most of the chapters, detailed as follows. Prerequisite material covering real numbers, intervals, increments, straight lines, distances, circles, and parabolas is presented in Appendices We reorganized and increased the number ofrelated rates examples, and we added new examples and exercises on graphing rational functions.
After carefully developing the integral concept, we turn our attention to its evaluation and connection to antiderivatives captured in the Fundamental Theorem of Calculus. The ensuing applications then define the various geometric ideas ofarea, volume, lengths ofpaths, and centroids all as limits of Riemann sums giving definite integrals, which can be evaluated by finding an antiderivative ofthe integrand.
We return later to the topic of solving more complicated first-order differential equations, after we derme and establish the transcendental functions and their properties. Although we do cover solutions to separable differential equations when treating exponential growth and decay applications in the chapter on transcendental functions, we organize the bulk of our material into two chapters which may be omitted for the calculus sequence.
We give an introductory treatment of first-order differential equations in Chapter 9, including a new section on systems and phase planes, with applications to the competitive-hunter and predator-prey models. We have added several new figures and exercises to the various sections, and we revised some of the proofs related to convergence ofpower series in order to improve the accessibility of the material for students.
We have done this, realizing that many departments choose to cover these topics at the beginning ofCalculus m, in preparation for their coverage ofvectors and multivariable calculus. Do you like this book?
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