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13. en evaluate the double integral in two ways. xdA, xdA, D is enclosed by the lines y = x, y = 0, x = 1

13. en evaluate the double integral in two ways. xdA, xdA, D is enclosed by the lines y = x, y = 0, x = 1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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16. 1008 14. CHAPTER 15 Multiple Integrals 15.2 EXERCISES 1-6 Evaluate the iterated integral. 1. (8x - 2y) dy dx 5. 3. fxe'dx dy D f. f* cos(s³) dt ds 2. f²f²x²y dx dy 4. x sin y dy dx 6. 12. Draw an example of a region that is (a) both type I and type II (b) neither type I nor type II niyatin TT/2 7-10 Evaluate the double integral. y 7. SS dA, D = {(x, y) | 0≤x≤ 4,0 ≤ y ≤ √√x} x² + 1 nobiniyo bilos 11. Draw an example of a region that is (a) type I but not type II (b) type II but not type I So So √I + eº dw dv 8. ff (2x + y) dA, D = {(x, y) | 1 ≤ y ≤ 2, y - 1<x< 1} D 9. fe²dA, D = {(x, y) | 0 ≤ y ≤ 3,0 ≤ x ≤ y} < < 10. ff y √x² − y² dA, D = {(x, y) | 0 ≤ x ≤ 2,0 ≤ y ≤x} - (3)A=1 13-14 Express D as a region of type I and also as a region of type II. Then evaluate the double integral in two ways. (0)Am 13. fx dA, D is enclosed by the lines y = x, y = 0, x = 1 D 15. y dA, D is bounded by y = x - 2, x = y² SS xy dA, D is enclosed by the curves y = x², y = 3x D Sveds. It 15-16 Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order and explain why it's easier. If y²e dA, y²exy dA, D is bounded by y = x, y = 4, x = 0 17-22 Evaluate the double integral. 17. x cos y dA, D is bounded by y = 0, y = x², x = 1 If x (x² + 2y) dA, D is bounded by y = x, y = x³, x ≥ 0 SS 19. ff y² dA, D is the triangular region with vertices (0, 1), (1, 2), (4,1) 18. 20. ff son og som er 0,01 1011 sayt xy dA, D is enclosed by the quarter-circle y = √1 = x², x = 0, and the axes 22. 21. ff (2x - y) dA, D sw. de intent 1 a, mite ID is bounded by the circle with center the origin and radius 2 23-32 Find the volume of the given solid. 23. Under the plane 3x + 2y - z = 0 and above the region achup BON ydw enclosed by the parabolas y = x² and x = A omuloy es zor 12 = y² on ozoftw 24. Under the surface z = 1 + x²y² and above the region enclosed by x = y² and x = 4 mo A 01951 & 19vo ff y dA, D is the triangular region with vertices (0, 0), (1, 1), and (4, 0) nidrige 25. Under the surface z = xy and above the triangle with vertices (1, 1), (4, 1), and (1, 2) 26. Enclosed by the paraboloid z = x² + y² + 1 and the planes (x = 0, y = 0, z = 0, and x + y = 2 27. The tetrahedron enclosed by the coordinate planes and the plane 2x + y + z = 4 28. Bounded by the planes z = x, y = x, x + y = 2, and z = 0 Samber bas night 30. Bounded by the cylinder y² + z² = 4 and the planes x = 2y, 1 x = 0, z = 0 in the first octant 29. Enclosed by the cylinders z = x²2, y = x² and the planes z = 0, y = 4 31. Bounded by the cylinder x² + y² x = 0, z = 0 in the first octant 32. Bounded by the cylinders x² + y² = r² and y² + z² = r² y² = 1 and the planes y = Z, 33. Use a graphing calculator or computer to estimate the x-coordinates of the points of intersection of the curves y = x² and y curves, estimate ff, x dA. 3x - x². If D is the region bounded by these =
Transcribed Image Text:16. 1008 14. CHAPTER 15 Multiple Integrals 15.2 EXERCISES 1-6 Evaluate the iterated integral. 1. (8x - 2y) dy dx 5. 3. fxe'dx dy D f. f* cos(s³) dt ds 2. f²f²x²y dx dy 4. x sin y dy dx 6. 12. Draw an example of a region that is (a) both type I and type II (b) neither type I nor type II niyatin TT/2 7-10 Evaluate the double integral. y 7. SS dA, D = {(x, y) | 0≤x≤ 4,0 ≤ y ≤ √√x} x² + 1 nobiniyo bilos 11. Draw an example of a region that is (a) type I but not type II (b) type II but not type I So So √I + eº dw dv 8. ff (2x + y) dA, D = {(x, y) | 1 ≤ y ≤ 2, y - 1<x< 1} D 9. fe²dA, D = {(x, y) | 0 ≤ y ≤ 3,0 ≤ x ≤ y} < < 10. ff y √x² − y² dA, D = {(x, y) | 0 ≤ x ≤ 2,0 ≤ y ≤x} - (3)A=1 13-14 Express D as a region of type I and also as a region of type II. Then evaluate the double integral in two ways. (0)Am 13. fx dA, D is enclosed by the lines y = x, y = 0, x = 1 D 15. y dA, D is bounded by y = x - 2, x = y² SS xy dA, D is enclosed by the curves y = x², y = 3x D Sveds. It 15-16 Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order and explain why it's easier. If y²e dA, y²exy dA, D is bounded by y = x, y = 4, x = 0 17-22 Evaluate the double integral. 17. x cos y dA, D is bounded by y = 0, y = x², x = 1 If x (x² + 2y) dA, D is bounded by y = x, y = x³, x ≥ 0 SS 19. ff y² dA, D is the triangular region with vertices (0, 1), (1, 2), (4,1) 18. 20. ff son og som er 0,01 1011 sayt xy dA, D is enclosed by the quarter-circle y = √1 = x², x = 0, and the axes 22. 21. ff (2x - y) dA, D sw. de intent 1 a, mite ID is bounded by the circle with center the origin and radius 2 23-32 Find the volume of the given solid. 23. Under the plane 3x + 2y - z = 0 and above the region achup BON ydw enclosed by the parabolas y = x² and x = A omuloy es zor 12 = y² on ozoftw 24. Under the surface z = 1 + x²y² and above the region enclosed by x = y² and x = 4 mo A 01951 & 19vo ff y dA, D is the triangular region with vertices (0, 0), (1, 1), and (4, 0) nidrige 25. Under the surface z = xy and above the triangle with vertices (1, 1), (4, 1), and (1, 2) 26. Enclosed by the paraboloid z = x² + y² + 1 and the planes (x = 0, y = 0, z = 0, and x + y = 2 27. The tetrahedron enclosed by the coordinate planes and the plane 2x + y + z = 4 28. Bounded by the planes z = x, y = x, x + y = 2, and z = 0 Samber bas night 30. Bounded by the cylinder y² + z² = 4 and the planes x = 2y, 1 x = 0, z = 0 in the first octant 29. Enclosed by the cylinders z = x²2, y = x² and the planes z = 0, y = 4 31. Bounded by the cylinder x² + y² x = 0, z = 0 in the first octant 32. Bounded by the cylinders x² + y² = r² and y² + z² = r² y² = 1 and the planes y = Z, 33. Use a graphing calculator or computer to estimate the x-coordinates of the points of intersection of the curves y = x² and y curves, estimate ff, x dA. 3x - x². If D is the region bounded by these =
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