Essential Calculus: Early Transcendentals
Essential Calculus: Early Transcendentals
2nd Edition
ISBN: 9781133112280
Author: James Stewart
Publisher: Cengage Learning
Question
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Chapter 12.1, Problem 46E

(a)

To determine

To explain: The similarity of the Fubini and Clairaut theorems.

(b)

To determine

To show: gxy=gyx=f(x,y)

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Chapter 12.1, Problem 44E
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Chapter 12 Solutions

Essential Calculus: Early Transcendentals

Ch. 12.1 - (a) Estimate the volume of the solid that lies...Ch. 12.1 - If R = [0, 4] [1, 2], use a Riemann sum with m =...Ch. 12.1 - (a) Use a Riemann sum with m = n = 2 to estimate...Ch. 12.1 - (a) Estimate the volume of the solid that lies...Ch. 12.1 - A 20-ft-by-30-ft swimming pool is filled with...Ch. 12.1 - A contour map is shown for a function f on the...Ch. 12.1 - 79 Evaluate the double integral by first...Ch. 12.1 - 7-9 Evaluate the double integral by first...Ch. 12.1 - Evaluate the double integral by first identifying...Ch. 12.1 - The integral R9y2dA, where R = [0, 4] [0, 2],...
Ch. 12.1 - Calculate the iterated integral. 15....Ch. 12.1 - Calculate the iterated integral. 12....Ch. 12.1 - 1120 Calculate the iterated integral. 13....Ch. 12.1 - 1120 Calculate the iterated integral. 16....Ch. 12.1 - Calculate the iterated integral. 19....Ch. 12.1 - Calculate the iterated integral. 20. 1315lnyxydydxCh. 12.1 - Calculate the iterated integral. 21....Ch. 12.1 - Calculate the iterated integral. 24....Ch. 12.1 - Calculate the iterated integral. 25....Ch. 12.1 - Calculate the iterated integral. 26. 0101s+tdsdtCh. 12.1 - Calculate the double integral. 28....Ch. 12.1 - Calculate the double integral. 29....Ch. 12.1 - Calculate the double integral. 31....Ch. 12.1 - Prob. 26ECh. 12.1 - Calculate the double integral. 33....Ch. 12.1 - Calculate the double integral. 24....Ch. 12.1 - Sketch the solid whose volume is given by the...Ch. 12.1 - Sketch the solid whose volume is given by the...Ch. 12.1 - Find the volume of the solid that lies under the...Ch. 12.1 - Find the volume of the solid that lies under the...Ch. 12.1 - Find the volume of the solid lying under the...Ch. 12.1 - Find the volume of the solid enclosed by the...Ch. 12.1 - Find the volume of the solid enclosed by the...Ch. 12.1 - Find the volume of the solid in the first octant...Ch. 12.1 - Find the volume of the solid enclosed by the...Ch. 12.1 - Graph the solid that lies between the surface z =...Ch. 12.1 - Find the average value of f over the given...Ch. 12.1 - Find the average value of f over the given...Ch. 12.1 - If f is a constant function, f(x, y) = k, and R =...Ch. 12.1 - Use the result of Exercise 41 to show that...Ch. 12.1 - Use symmetry to evaluate the double integral. 49....Ch. 12.1 - Use symmetry to evaluate the double integral. 50....Ch. 12.1 - Prob. 46ECh. 12.2 - 16 Evaluate the iterated integral. 1. 040yxy2dxdyCh. 12.2 - Evaluate the iterated integral. 2. 012x2(xy)dydxCh. 12.2 - 16 Evaluate the iterated integral. 3....Ch. 12.2 - Evaluate the iterated integral. 2. 02y2yxydxdyCh. 12.2 - Evaluate the iterated integral. 5....Ch. 12.2 - Evaluate the iterated integral. 6. 010ex1+exdwdvCh. 12.2 - 710 Evaluate the double integral. 7....Ch. 12.2 - Evaluate the double integral. 8....Ch. 12.2 - 710 Evaluate the double integral. 9....Ch. 12.2 - Evaluate the double integral. 10....Ch. 12.2 - Express D as a region of type I and also as a...Ch. 12.2 - Express D as a region of type I and also as a...Ch. 12.2 - Set up iterated integrals for both orders of...Ch. 12.2 - Set up iterated integrals for both orders of...Ch. 12.2 - Evaluate the double integral. 17.DxcosydA, D is...Ch. 12.2 - Evaluate the double integral. 18. D(x2+2y)dA, D is...Ch. 12.2 - Evaluate the double integral. 19. Dy2dA, D is the...Ch. 12.2 - Evaluate the double integral. 18....Ch. 12.2 - Prob. 19ECh. 12.2 - 1520 Evaluate the double integral. 20. D2xydA, D...Ch. 12.2 - 2130 Find the volume of the given solid. 21. Under...Ch. 12.2 - Prob. 22ECh. 12.2 - Prob. 23ECh. 12.2 - Prob. 24ECh. 12.2 - 2130 Find the volume of the given solid. 25....Ch. 12.2 - Find the volume of the given solid. 28. Bounded by...Ch. 12.2 - Find the volume of the given solid. 29. Enclosed...Ch. 12.2 - Find the volume of the given solid. 30. Bounded by...Ch. 12.2 - Find the volume of the given solid. 31. Bounded by...Ch. 12.2 - Prob. 30ECh. 12.2 - Prob. 31ECh. 12.2 - Prob. 32ECh. 12.2 - Sketch the solid whose volume is given by the...Ch. 12.2 - Sketch the solid whose volume is given by the...Ch. 12.2 - Sketch the region of integration and change the...Ch. 12.2 - Sketch the region of integration and change the...Ch. 12.2 - Sketch the region of integration and change the...Ch. 12.2 - Sketch the region of integration and change the...Ch. 12.2 - Sketch the region of integration and change the...Ch. 12.2 - Prob. 42ECh. 12.2 - Evaluate the integral by reversing the order of...Ch. 12.2 - 43-48 Evaluate the integral by reversing the order...Ch. 12.2 - 4348 Evaluate the integral by reversing the order...Ch. 12.2 - Prob. 46ECh. 12.2 - Evaluate the integral by reversing the order of...Ch. 12.2 - Evaluate the integral by reversing the order of...Ch. 12.2 - Express D as a union of regions of type I or type...Ch. 12.2 - Express D as a union of regions of type I or type...Ch. 12.2 - 5152 Use Property 11 to estimate the value of the...Ch. 12.2 - Use Property 11 to estimate the value of the...Ch. 12.2 - Prove Property 11.Ch. 12.2 - In evaluating a double integral over a region D, a...Ch. 12.2 - Prob. 55ECh. 12.2 - Prob. 56ECh. 12.2 - Prob. 57ECh. 12.2 - Prob. 58ECh. 12.2 - Prob. 59ECh. 12.3 - 14 A region R is shown. Decide whether to use...Ch. 12.3 - Prob. 2ECh. 12.3 - Prob. 3ECh. 12.3 - Prob. 4ECh. 12.3 - Sketch the region whose area is given by the...Ch. 12.3 - Prob. 6ECh. 12.3 - Evaluate the given integral by changing to polar...Ch. 12.3 - Prob. 8ECh. 12.3 - Evaluate the given integral by changing to polar...Ch. 12.3 - Prob. 10ECh. 12.3 - Prob. 12ECh. 12.3 - Prob. 11ECh. 12.3 - Use a double integral to find the area of the...Ch. 12.3 - Use a double integral to find the area of the...Ch. 12.3 - Prob. 13ECh. 12.3 - Prob. 14ECh. 12.3 - Use polar coordinates to find the volume of the...Ch. 12.3 - Prob. 15ECh. 12.3 - Use polar coordinates to find the volume of the...Ch. 12.3 - 1319 Use polar coordinates to find the volume of...Ch. 12.3 - Use polar coordinates to find the volume of the...Ch. 12.3 - (a) A cylindrical drill with radius r1 is used to...Ch. 12.3 - 2326 Evaluate the iterated integral by converting...Ch. 12.3 - Evaluate the iterated integral by converting to...Ch. 12.3 - 2326 Evaluate the iterated integral by converting...Ch. 12.3 - Evaluate the iterated integral by converting to...Ch. 12.3 - A swimming pool is circular with a 40-ft diameter....Ch. 12.3 - An agricultural sprinkler distributes water in a...Ch. 12.3 - Use polar coordinates to combine the sum...Ch. 12.3 - (a) We define the improper integral (over the...Ch. 12.3 - Use the result of Exercise 30 part (c) to evaluate...Ch. 12.4 - Electric charge is distributed over the rectangle...Ch. 12.4 - Electric charge is distributed over the disk x2 +...Ch. 12.4 - Find the mass and center of mass of the lamina...Ch. 12.4 - Find the mass and center of mass of the lamina...Ch. 12.4 - Find the mass and center of mass of the lamina...Ch. 12.4 - 3-10 Find the mass and center of mass of the...Ch. 12.4 - Find the mass and center of mass of the lamina...Ch. 12.4 - 3-10 Find the mass and center of mass of the...Ch. 12.4 - 310 Find the mass and center of mass of the lamina...Ch. 12.4 - 3-10 Find the mass and center of mass of the...Ch. 12.4 - A lamina occupies the part of the disk x2 + y2 1...Ch. 12.4 - Find the center of mass of the lamina in Exercise...Ch. 12.4 - The boundary of a lamina consists of the...Ch. 12.4 - Find the center of mass of the lamina in Exercise...Ch. 12.4 - Find the center of mass of a lamina in the shape...Ch. 12.4 - A lamina occupies the region inside the circle x2...Ch. 12.4 - Find the moments of inertia Ix, Iy, I0 for the...Ch. 12.4 - Find the moments of inertia Ix, Iy, I0 for the...Ch. 12.4 - Find the moments of inertia Ix, Iy, lo for the...Ch. 12.4 - Consider a square fan blade with sides of length 2...Ch. 12.4 - A lamina with constant density (x, y) = occupies...Ch. 12.4 - A lamina with constant density (x, y) = occupies...Ch. 12.5 - Evaluate the integral in Example 1, integrating...Ch. 12.5 - Evaluate the integral E(xy+z2)dv, where...Ch. 12.5 - Evaluate the iterated integral....Ch. 12.5 - 36 Evaluate the iterated integral. 5....Ch. 12.5 - 00x0xzx2sinydydzdxCh. 12.5 - Evaluate the iterated integral. 6....Ch. 12.5 - Evaluate the triple integral. 9. EydV, where...Ch. 12.5 - Evaluate the triple integral. 10.EezydV, where...Ch. 12.5 - Evaluate the triple integral. 11. Ezx2+z2dV, where...Ch. 12.5 - Evaluate the triple integral. 12. EsinydV, where E...Ch. 12.5 - Evaluate the triple integral. 13. E6xydV, where E...Ch. 12.5 - Prob. 12ECh. 12.5 - 716 Evaluate the triple integral. 13. T x2 dV,...Ch. 12.5 - 7-16 Evaluate the triple integral. 14. TxyzdV,...Ch. 12.5 - Evaluate the triple integral. 17. ExdV, where E is...Ch. 12.5 - Evaluate the triple integral. 18. EzdV, where E is...Ch. 12.5 - Prob. 17ECh. 12.5 - Use a triple integral to find the volume of the...Ch. 12.5 - Use a triple integral to find the volume of the...Ch. 12.5 - Use a triple integral to find the volume of the...Ch. 12.5 - Prob. 23ECh. 12.5 - Prob. 24ECh. 12.5 - Prob. 25ECh. 12.5 - Prob. 26ECh. 12.5 - Express the integralEf(x,y,z)dV, as an iterated...Ch. 12.5 - Prob. 28ECh. 12.5 - Prob. 29ECh. 12.5 - Prob. 30ECh. 12.5 - Prob. 31ECh. 12.5 - Prob. 32ECh. 12.5 - Write five other iterated integrals that are equal...Ch. 12.5 - Prob. 34ECh. 12.5 - Prob. 35ECh. 12.5 - Prob. 36ECh. 12.5 - 3740 Find the mass and center of mass of the solid...Ch. 12.5 - Prob. 38ECh. 12.5 - Prob. 39ECh. 12.5 - Prob. 40ECh. 12.5 - Prob. 45ECh. 12.5 - Prob. 46ECh. 12.5 - Prob. 47ECh. 12.5 - Prob. 48ECh. 12.5 - Prob. 41ECh. 12.5 - Prob. 42ECh. 12.5 - Prob. 44ECh. 12.5 - Prob. 49ECh. 12.5 - Prob. 50ECh. 12.6 - Plot the point whose cylindrical coordinates are...Ch. 12.6 - Prob. 2ECh. 12.6 - Prob. 3ECh. 12.6 - Prob. 4ECh. 12.6 - Prob. 5ECh. 12.6 - Prob. 6ECh. 12.6 - 78 Identify the surface whose equation is given....Ch. 12.6 - Prob. 8ECh. 12.6 - Prob. 9ECh. 12.6 - Prob. 10ECh. 12.6 - Prob. 11ECh. 12.6 - Prob. 12ECh. 12.6 - Prob. 13ECh. 12.6 - Prob. 14ECh. 12.6 - Sketch the solid whose volume is given by the...Ch. 12.6 - Sketch the solid whose volume is given by the...Ch. 12.6 - Use cylindrical coordinates. 17. Evaluate...Ch. 12.6 - Prob. 18ECh. 12.6 - Prob. 19ECh. 12.6 - 21-32 Use spherical coordinates. 20. Evaluate...Ch. 12.6 - Use cylindrical coordinates. 21. Evaluate Ex2dV,...Ch. 12.6 - Prob. 22ECh. 12.6 - Use cylindrical coordinates. 23. Find the volume...Ch. 12.6 - Prob. 24ECh. 12.6 - 1728 Use cylindrical coordinates. 25. (a) Find the...Ch. 12.6 - Use cylindrical coordinates. 26. (a) Find the...Ch. 12.6 - Use cylindrical coordinates. 27. Find the mass and...Ch. 12.6 - Use cylindrical coordinates. 28. Find the mass of...Ch. 12.6 - Evaluate the integral by changing to cylindrical...Ch. 12.6 - Prob. 30ECh. 12.6 - Prob. 31ECh. 12.7 - Prob. 1ECh. 12.7 - Prob. 2ECh. 12.7 - Prob. 3ECh. 12.7 - Prob. 4ECh. 12.7 - Prob. 5ECh. 12.7 - Prob. 6ECh. 12.7 - 78 Identify the surface whose equation is given....Ch. 12.7 - Identify the surface whose equation is given. 8. ...Ch. 12.7 - Prob. 9ECh. 12.7 - Prob. 10ECh. 12.7 - 1114 Sketch the solid described by the given...Ch. 12.7 - Sketch the solid described by the given...Ch. 12.7 - 1112 Sketch the solid described by the given...Ch. 12.7 - Sketch the solid described by the given...Ch. 12.7 - A solid lies above the cone z = x2+y2 and below...Ch. 12.7 - Prob. 16ECh. 12.7 - Prob. 17ECh. 12.7 - Sketch the solid whose volume is given by the...Ch. 12.7 - Prob. 19ECh. 12.7 - Prob. 20ECh. 12.7 - Use spherical coordinates. 21. Evaluate B (x2+y2 +...Ch. 12.7 - 21-32 Use spherical coordinates. 22. Evaluate...Ch. 12.7 - Prob. 23ECh. 12.7 - 21-32 Use spherical coordinates. 24. Evaluate...Ch. 12.7 - Use spherical coordinates. 25. Evaluate E xe x2 +...Ch. 12.7 - Prob. 26ECh. 12.7 - Use spherical coordinates. 29. (a) Find the volume...Ch. 12.7 - Use spherical coordinates. 30. Find the volume of...Ch. 12.7 - Prob. 29ECh. 12.7 - Use spherical coordinates. 32. Let H be a solid...Ch. 12.7 - Prob. 31ECh. 12.7 - Use spherical coordinates. 34. Find the mass and...Ch. 12.7 - Use cylindrical or spherical coordinates,...Ch. 12.7 - Use cylindrical or spherical coordinates,...Ch. 12.7 - Evaluate the integral by changing to spherical...Ch. 12.7 - Evaluate the integral by changing to spherical...Ch. 12.7 - Evaluate the integral by changing to spherical...Ch. 12.7 - A model for the density of the earths atmosphere...Ch. 12.7 - Use a graphing device to draw a silo consisting of...Ch. 12.7 - Prob. 42ECh. 12.7 - Show that x2+y2+z2e-(x2+y2+z2) dx dy dz = 2 (The...Ch. 12.7 - Prob. 45ECh. 12.8 - 16 Find the Jacobian of the transformation. 1. x =...Ch. 12.8 - Find the Jacobian of the transformation. 2. x =...Ch. 12.8 - 16 Find the Jacobian of the transformation. 3. x =...Ch. 12.8 - Find the Jacobian of the transformation. 4. x =...Ch. 12.8 - 16 Find the Jacobian of the transformation. 5. x =...Ch. 12.8 - Find the Jacobian of the transformation. 6. x = v...Ch. 12.8 - Find the image of the set S under the given...Ch. 12.8 - Find the image of the set S under the given...Ch. 12.8 - Find the image of the set S under the given...Ch. 12.8 - Find the image of the set S under the given...Ch. 12.8 - A region R in the xy-plane is given. Find...Ch. 12.8 - Prob. 12ECh. 12.8 - A region R in the xy-plane is given. Find...Ch. 12.8 - A region R in the xy-plane is given. Find...Ch. 12.8 - Use the given transformation to evaluate the...Ch. 12.8 - Use the given transformation to evaluate the...Ch. 12.8 - Use the given transformation to evaluate the...Ch. 12.8 - Use the given transformation to evaluate the...Ch. 12.8 - Use the given transformation to evaluate the...Ch. 12.8 - Use the given transformation to evaluate the...Ch. 12.8 - (a) Evaluate E dV, where E is the solid enclosed...Ch. 12.8 - An important problem in thermodynamics is to find...Ch. 12.8 - Evaluate the integral by making an appropriate...Ch. 12.8 - Evaluate the integral by making an appropriate...Ch. 12.8 - Evaluate the integral by making an appropriate...Ch. 12.8 - Evaluate the integral by making an appropriate...Ch. 12.8 - Evaluate the integral by making an appropriate...Ch. 12.8 - Let f be continuous oil [0, 1] and letRbe the...Ch. 12 - Prob. 1RCCCh. 12 - Prob. 2RCCCh. 12 - Prob. 3RCCCh. 12 - Prob. 4RCCCh. 12 - Prob. 7RCCCh. 12 - Prob. 5RCCCh. 12 - Suppose a solid object occupies the region E and...Ch. 12 - Prob. 8RCCCh. 12 - (a) If a transformation T is given by x = g(u, v),...Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - Determine whether the statement is true or false....Ch. 12 - A contour map is shown for a function f on the...Ch. 12 - Use the Midpoint Rule to estimate the integral in...Ch. 12 - Calculate the iterated integral. 3....Ch. 12 - Calculate the iterated integral. 4. 0101yexydxdyCh. 12 - Calculate the iterated integral. 5....Ch. 12 - Calculate the iterated integral. 6. 01xex3xy2dydxCh. 12 - Calculate the iterated integral. 7....Ch. 12 - Calculate the iterated integral. 8....Ch. 12 - Write Rf(x,y)dA as an iterated integral, where R...Ch. 12 - Write Rf(x,y)dA as an iterated integral, where R...Ch. 12 - Prob. 39RECh. 12 - Prob. 40RECh. 12 - Prob. 41RECh. 12 - Prob. 42RECh. 12 - Prob. 43RECh. 12 - Prob. 44RECh. 12 - Describe the region whose area is given by the...Ch. 12 - Prob. 12RECh. 12 - Prob. 13RECh. 12 - Prob. 14RECh. 12 - Prob. 15RECh. 12 - Prob. 16RECh. 12 - Prob. 17RECh. 12 - Prob. 18RECh. 12 - Prob. 19RECh. 12 - Prob. 20RECh. 12 - Prob. 21RECh. 12 - Prob. 22RECh. 12 - Prob. 23RECh. 12 - Prob. 24RECh. 12 - Prob. 25RECh. 12 - Prob. 26RECh. 12 - Prob. 27RECh. 12 - Prob. 28RECh. 12 - Prob. 29RECh. 12 - Prob. 30RECh. 12 - Prob. 31RECh. 12 - Prob. 32RECh. 12 - Prob. 33RECh. 12 - Prob. 34RECh. 12 - Prob. 35RECh. 12 - Prob. 36RECh. 12 - Prob. 37RECh. 12 - Use polar coordinates to evaluate...Ch. 12 - Use spherical coordinates to evaluate...Ch. 12 - Rewrite the integral 11x2101yf(x,y,z)dzdydxas an...Ch. 12 - Prob. 48RECh. 12 - Use the transformation u = x y, v = x + y to...Ch. 12 - Use the transformation x = u2, y = v2 z = w2 to...Ch. 12 - Use the change of variables formula and an...Ch. 12 - Prob. 52RE
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