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19. The tetrahedron enclosed by the coordinate planes and the plane 2x + y + z = 4

19. The tetrahedron enclosed by the coordinate planes and the plane 2x + y + z = 4

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
Chapter7: Locus And Concurrence
Section7.2: Concurrence Of Lines
Problem 7E: Which lines or line segments or rays must be drawn or constructed in a triangle to locate its a...
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CAS CAS 1038 CHAPTER 15 Multiple Integrals 16. f, xz dv, where T is the solid tetrahedron with vertices (0, 0, 0), (1, 0, 1), (0, 1, 1), and (0, 0, 1) 17. Ex dV, where E is bounded by the paraboloid x = 4y² + 4z² and the plane x = 4 18. Ez dv, where E is bounded by the cylinder y2 + z² = 9 and the planes x = 0, y = 3x, and z = 0 in the first octant 19-22 Use a triple integral to find the volume of the given solid. 19. The tetrahedron enclosed by the coordinate planes and the plane 2x + y + z = 4 20. The solid enclosed by the paraboloids y = x² + z² and y = 8 - x² = z² 21. The solid enclosed by the cylinder y = z = 0 and y + z = 1 x² and the planes 22. The solid enclosed by the cylinder x² + z² = 4 and the planes y = 1 and y + z = 4 1 4 and the 23. (a) Express the volume of the wedge in the first octant that is cut from the cylinder y² + z² = 1 by the planes y = x and x = 1 as a triple integral. (b) Use either the Table of Integrals (on Reference Pages 6-10) or a computer algebra system to find the exact value of the triple integral in part (a). 24. (a) In the Midpoint Rule for triple integrals we use a triple Riemann sum to approximate a triple integral over a box B, where f(x, y, z) is evaluated at the center (Xi, yi, Zk) of the box Bijk. Use the Midpoint Rule to estimate fff √√x² + y² + z² dV, where B is the cube defined by 0 ≤ x ≤ 4, 0≤ y ≤ 4, 0≤z≤ 4. Divide B into eight cubes of equal size. (b) Use a computer algebra system to approximate the integral in part (a) correct to the nearest integer. Com- pare with the answer to part (a). 25-26 Use the Midpoint Rule for triple integrals (Exercise 24) to estimate the value of the integral. Divide B into eight sub- boxes of equal size. 25. fff cos(xyz) dV, where B = {(x, y, z) | 0≤x≤ 1, 0 ≤ y ≤ 1,0 ≤ z ≤ 1} 26. B √xe dv, where dod nad Vb B = {(x, y, z) | 0≤x≤ 4, 0 ≤ y ≤ 1,0 ≤ z ≤ 2} 10 T 2-2z 27. f * ²² dy dz dx Sox Jo List 27-28 Sketch the solid whose volume is given by the iterated integral. bra 28. ffdx dz dy 29-32 Express the integral JJJE f(x, y, z) dV as an iterated integral in six different ways, where E is the solid bounded by the given surfaces. 29. y = 4x² - 4z², y = 0 30. y² + z² = 9, x = -2, x = 2 31. y = x², z = 0, y + 2z = 4 32. x = 2, y = 2, z = 0, x + y2z = 2 33. The figure shows the region of integration for the integral So S Sof(x, y, z) dz dy dx M Rewrite this integral as an equivalent iterated integral in five other orders. shxis oris anotant XX 35. (8P13 ZA 1- y=√√√x- 34. The figure shows the region of integration for the inte (1-x² (1-x So f*²1* f(x, y, z) dy dz dx JO z=1-x² 1 0 Rewrite this integral as an equivalent iterated integral five other orders. ZA 1 z=1-y 0 1 y So f' f f(x, y, z) dz dx dy 36. f' f f f(x, y, z) dx dz dy y = 1- x 35-36 Write five other iterated integrals that are equal given iterated integral. 1 y
Transcribed Image Text:CAS CAS 1038 CHAPTER 15 Multiple Integrals 16. f, xz dv, where T is the solid tetrahedron with vertices (0, 0, 0), (1, 0, 1), (0, 1, 1), and (0, 0, 1) 17. Ex dV, where E is bounded by the paraboloid x = 4y² + 4z² and the plane x = 4 18. Ez dv, where E is bounded by the cylinder y2 + z² = 9 and the planes x = 0, y = 3x, and z = 0 in the first octant 19-22 Use a triple integral to find the volume of the given solid. 19. The tetrahedron enclosed by the coordinate planes and the plane 2x + y + z = 4 20. The solid enclosed by the paraboloids y = x² + z² and y = 8 - x² = z² 21. The solid enclosed by the cylinder y = z = 0 and y + z = 1 x² and the planes 22. The solid enclosed by the cylinder x² + z² = 4 and the planes y = 1 and y + z = 4 1 4 and the 23. (a) Express the volume of the wedge in the first octant that is cut from the cylinder y² + z² = 1 by the planes y = x and x = 1 as a triple integral. (b) Use either the Table of Integrals (on Reference Pages 6-10) or a computer algebra system to find the exact value of the triple integral in part (a). 24. (a) In the Midpoint Rule for triple integrals we use a triple Riemann sum to approximate a triple integral over a box B, where f(x, y, z) is evaluated at the center (Xi, yi, Zk) of the box Bijk. Use the Midpoint Rule to estimate fff √√x² + y² + z² dV, where B is the cube defined by 0 ≤ x ≤ 4, 0≤ y ≤ 4, 0≤z≤ 4. Divide B into eight cubes of equal size. (b) Use a computer algebra system to approximate the integral in part (a) correct to the nearest integer. Com- pare with the answer to part (a). 25-26 Use the Midpoint Rule for triple integrals (Exercise 24) to estimate the value of the integral. Divide B into eight sub- boxes of equal size. 25. fff cos(xyz) dV, where B = {(x, y, z) | 0≤x≤ 1, 0 ≤ y ≤ 1,0 ≤ z ≤ 1} 26. B √xe dv, where dod nad Vb B = {(x, y, z) | 0≤x≤ 4, 0 ≤ y ≤ 1,0 ≤ z ≤ 2} 10 T 2-2z 27. f * ²² dy dz dx Sox Jo List 27-28 Sketch the solid whose volume is given by the iterated integral. bra 28. ffdx dz dy 29-32 Express the integral JJJE f(x, y, z) dV as an iterated integral in six different ways, where E is the solid bounded by the given surfaces. 29. y = 4x² - 4z², y = 0 30. y² + z² = 9, x = -2, x = 2 31. y = x², z = 0, y + 2z = 4 32. x = 2, y = 2, z = 0, x + y2z = 2 33. The figure shows the region of integration for the integral So S Sof(x, y, z) dz dy dx M Rewrite this integral as an equivalent iterated integral in five other orders. shxis oris anotant XX 35. (8P13 ZA 1- y=√√√x- 34. The figure shows the region of integration for the inte (1-x² (1-x So f*²1* f(x, y, z) dy dz dx JO z=1-x² 1 0 Rewrite this integral as an equivalent iterated integral five other orders. ZA 1 z=1-y 0 1 y So f' f f(x, y, z) dz dx dy 36. f' f f f(x, y, z) dx dz dy y = 1- x 35-36 Write five other iterated integrals that are equal given iterated integral. 1 y
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