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(Answers to Exercises for Chapter 6: Applications of Integrals) A.6.1 CHAPTER 6: APPLICATIONS OF INTEGRALS SECTION 6.1: AREA 1) a) i) ii) ± x 2 () ± x 2 ± 8 ² ³ ´ µ dx ± 2 2 , or 2 ± x 2 ± x 2 ± 8 ² ³ ´ µ dx 0 2 , or 4 ± x 2 ±± 4 ² ³ ´ µ dx 0 2 (by symmetry) iii) 2 y + 8 dy ± 8 ± 4 ² + 2 ± y dy ± 4 0 ² , or 2 y + 8 dy ± 8 ± 4 ² + 2 ± y dy ± 4 0 ² , or 4 ± y dy ± 4 0 ² (by symmetry) iv) 64 3 m 2 , or 21 1 3 m 2

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(Answers to Exercises for Chapter 6: Applications of Integrals) A.6.2 b) i) ii) 4 ± x ± 1 ± x 2 ² ³ ´ µ · dx ± 5 3 ¸ + 24 ± x dx 3 4 ¸ iii) 4 ± y 2 () ± 1 ± 2 y ² ³ ´ µ dy ± 1 3 iv) 32 3 m 2 2) m 2 . Hint: The setup is given by: cos t ± sin t dt 0 ² /4 ³ + sin t ± cos t dt + cos t ± sin t dt 5 2 ³ 5 ³ . 3) a) 343 24 m 2 , or 14 7 24 m 2 . Hint: Setup is: x 2 + 5 x ± 2 ± 3 x 2 ± 5 ² ³ ´ µ dx ± 1/2 3 . b) 61 3 m 2 , or 20 1 3 m 2 . Hint: Setup is: xx 2 + 16 dx 0 3 ± . c) 1 12 m 2 . Hint: Setup is: ± y ± 2 y 2 ± y 3 ² ³ ´ µ dy ± 1 0 .
(Answers to Exercises for Chapter 6: Applications of Integrals) A.6.3 SECTION 6.2: VOLUMES OF SOLIDS OF REVOLUTION – DISKS AND WASHERS 1) a) b) 2 ± 3 m 3 . Hint: Setup is: 2 ² x 2 ³ ´ µ · ¸ 2 dx 0 2 ¹ . c) 4 3 m 3 . Hint: Setup is: 2 ² 2 y () 2 dy 0 1 ³ . 2) a) 512 3 m 3 . Hint: Setup is: 0 ² x 2 ² 8 2 ² 0 ²² x 2 2 ³ ´ µ · ¸ dx ² 2 2 ¹ , or 2 0 ² x 2 ² 8 2 ² 0 x 2 2 ³ ´ µ · ¸ dx 0 2 ¹ by exploiting symmetry. b) 16 m 3 . Hint 1: Setup is: y + 8 dy ² 8 ² 4 ³ + ² y dy ² 4 0 ³ , or 2 ² y dy ² 4 0 ³ by exploiting symmetry. Hint 2: You only want to revolve half of the region 360 ± around the axis of revolution.

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