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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 48

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online