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# Chapter 7 - SECTION 7.1(PAGE 376 R A ADAMS CALCULUS CHAPTER...

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SECTION 7.1 (PAGE 376) R. A. ADAMS: CALCULUS CHAPTER 7. APPLICATIONS OF INTE- GRATION Section 7.1 Volumes of Solids of Revolution (page 376) 1. By slicing: V = π 1 0 x 4 dx = π 5 cu. units. By shells: V = 2 π 1 0 y ( 1 y ) dy = 2 π y 2 2 2 y 5 / 2 5 1 0 = π 5 cu. units. y x y = x 2 ( 1 , 1 ) x Fig. 7.1.1 2. Slicing: V = π 1 0 ( 1 y ) dy = π y 1 2 y 2 1 0 = π 2 cu . units . Shells: V = 2 π 1 0 x 3 dx = 2 π x 4 4 1 0 = π 2 cu . units . y x y = x 2 1 Fig. 7.1.2 3. By slicing: V = π 1 0 ( x x 4 ) dx = π x 2 2 x 5 5 1 0 = 3 π 10 cu. units. By shells: V = 2 π 1 0 y ( y y 2 ) dy = 2 π 2 y 5 / 2 5 y 4 4 1 0 = 3 π 10 cu. units. y x x y = x 2 y = x Fig. 7.1.3 4. Slicing: V = π 1 0 ( y y 4 ) dy = π 1 2 y 2 1 5 y 5 1 0 = 3 π 10 cu . units . Shells: V = 2 π 1 0 x ( x 1 / 2 x 2 ) dx = 2 π 2 5 x 5 / 2 1 4 x 4 1 0 = 3 π 10 cu . units . y x y = x y = x 2 ( 1 , 1 ) Fig. 7.1.4 264

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INSTRUCTOR’S SOLUTIONS MANUAL SECTION 7.1 (PAGE 376) 5. a) About the x -axis: V = π 2 0 x 2 ( 2 x ) 2 dx = π 2 0 ( 4 x 2 4 x 3 + x 4 ) dx = π 4 x 3 3 x 4 + x 5 5 2 0 = 16 π 15 cu. units. b) About the y -axis: V = 2 π 2 0 x 2 ( 2 x ) dy = 2 π 2 x 3 3 x 4 4 2 0 = 8 π 3 cu. units. y = 2 x x 2 2 y = 2 x x 2 2 y y x x (b) (a) Fig. 7.1.5 6. Rotate about a) the x -axis V = π 1 0 ( x 2 x 4 ) dx = π 1 3 x 3 1 5 x 5 1 0 = 2 π 15 cu . units . b) the y -axis V = 2 π 1 0 x ( x x 2 ) dx = 2 π 1 3 x 3 1 4 x 4 1 0 = π 6 cu . units . y x ( 1 , 1 ) y = x 2 y = x Fig. 7.1.6 7. a) About the x -axis: V = 2 π 3 0 y ( 4 y y 2 y ) dy = 2 π y 3 y 4 4 3 0 = 27 π 2 cu. units. b) About the y -axis: V = π 3 0 ( 4 y y 2 ) 2 y 2 dy = π 3 0 ( 15 y 2 8 y 3 + y 4 ) dy = π 5 y 3 2 y 4 + y 5 5 3 0 = 108 π 5 cu. units. y x ( 3 , 3 ) x = 4 y y 2 x = y Fig. 7.1.7 8. Rotate about a) the x -axis V = π π 0 [ ( 1 + sin x ) 2 1] dx = π π 0 ( 2 sin x + sin 2 x ) dx = 2 π cos x + π 2 x π 4 sin 2 x π 0 = 4 π + 1 2 π 2 cu . units . 265
SECTION 7.1 (PAGE 376) R. A. ADAMS: CALCULUS b) the y -axis V = 2 π π 0 x sin x dx U = x dU = dx dV = sin x dx V = − cos x = 2 π x cos x π 0 + π 0 cos x dx = 2 π 2 cu . units . 9. a) About the x -axis: V = π 1 0 4 1 ( 1 + x 2 ) 2 dx Let x = tan θ dx = sec 2 θ d θ = 4 π π π/ 4 0 sec 2 θ sec 4 θ d θ = 4 π π π/ 4 0 cos 2 θ d θ = 4 π π 2 + sin θ cos θ) π/ 4 0 = 4 π π 2 8 π 4 = 15 π 4 π 2 8 cu. units. b) About the y -axis: V = 2 π 1 0 x 2 1 1 + x 2 dx = 2 π x 2 1 2 ln ( 1 + x 2 ) 1 0 = 2 π 1 1 2 ln 2 = 2 π π ln 2 cu. units. y x y = 1 1 + x 2 y = 2 x 1 Fig. 7.1.9 10. By symmetry, rotation about the x -axis gives the same volume as rotation about the y -axis, namely V = 2 π 3 1 / 3 x 10 3 x 1 x dx = 2 π 5 3 x 2 1 3 x 3 x 3 1 / 3 = 512 π 81 cu . units . y x 3 x + 3 y = 10 y = 1 x ( 3 , 1 / 3 ) ( 1 / 3 , 3 ) Fig. 7.1.10 11. V = 2 × 2 π 1 0 ( 2 x )( 1 x ) dx = 4 π 1 0 ( 2 3 x + x 2 ) dx = 4 π 2 x 3 x 2 2 + x 3 3 1 0 = 10 π 3 cu. units. y x y x + y = 1 x = 2 x Fig. 7.1.11 12. V = π 1 1 [ ( 1 ) 2 ( x 2 ) 2 ] dx = π x 1 5 x 5 1 1 = 8 π 5 cu . units . y x x 2 y = 1 y = 1 x 2 x dx Fig. 7.1.12 13. The volume remaining is V = 2 × 2 π 2 1 x 4 x 2 dx Let u = 4 x 2 du = − 2 x dx = 2 π 3 0 u du = 4 π 3 u 3 / 2 3 0 = 4 π 3 cu. units.

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Chapter 7 - SECTION 7.1(PAGE 376 R A ADAMS CALCULUS CHAPTER...

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