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

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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 ² y 2 2 2 y 5 / 2 5 ³ ´ ´ ´ ´ 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 ) = π µ y 1 2 y 2 ¶´ ´ ´ ´ 1 0 = π 2 cu . units . Shells: V = 2 π ± 1 0 x 3 = 2 π µ 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 ) = π ² x 2 2 x 5 5 ³ ´ ´ ´ ´ 1 0 = 3 π 10 cu. units. By shells: V = 2 π ± 1 0 y ( y y 2 ) = 2 π ² 2 y 5 / 2 5 y 4 4 ³ ´ ´ ´ ´ 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 ) = π µ 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 ) = 2 π µ 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 ) = π ² 4 x 3 3 x 4 + x 5 5 ³ ´ ´ ´ ´ 2 0 = 16 π 15 cu. units. b) About the y -axis: V = 2 π ± 2 0 x 2 ( 2 x ) dy = 2 π µ 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 ) = π µ 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 ) = 2 π µ 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 ) = 2 π µ 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 ¸ = π ± 3 0 ( 15 y 2 8 y 3 + y 4 ) = π ² 5 y 3 2 y 4 + y 5 5 ³ ´ ´ ´ ´ 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] = π ± π 0 ( 2 sin x + sin 2 x ) = µ 2 π cos x + π 2 x π 4 sin 2 x ¶´ ´ ´ ´ π 0 = 4 π + 1 2 π 2 cu . units . 265
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SECTION 7.1 (PAGE 376) R. A. ADAMS: CALCULUS b) the y -axis V = 2 π ± π 0 x sin xdx U = x dU = dx dV = sin V =− cos x = 2 π ² x cos x ³ ³ ³ ³ π 0 + ± π 0 cos ´ = 2 π 2 cu . units . 9. a) About the x -axis: V = π ± 1 0 µ 4 1 ( 1 + x 2 ) 2 Let x = tan θ = sec 2 θ d θ = 4 π π ± π/ 4 0 sec 2 θ sec 4 θ d θ = 4 π π ± 4 0 cos 2 θ d θ = 4 π π 2 + sin θ cos θ) ³ ³ ³ ³ 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 µ 2 1 1 + x 2 = 2 π µ x 2 1 2 ln ( 1 + x 2 ) ¶³ ³ ³ ³ 1 0 = 2 π µ 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 µ 10 3 x 1 x = 2 π µ 5 3 x 2 1 3 x 3 x ¶³ ³ ³ ³ 3 1 / 3 = 512 π 81 cu . units .
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This note was uploaded on 12/16/2009 for the course FEW, FEWEB 400567 taught by Professor Moerdersen during the Fall '09 term at Vrije Universiteit Amsterdam.

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

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