Calculus - Chapter 5 Applications of the Definite Integral...

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Chapter 5 Applications of the Definite Integral 153 Chapter 5 Section 5.1 5. 3 2 , 1, 1 3 y x y x x = = ( ) 3 3 2 1 3 4 3 1 1 4 3 81 27 1 1 3 1 4 3 4 3 160 12 40 3 x x dx x x x = + = + + = = 7. , 1, 2 0 x y e y x x = = 0 2 0 2 2 2 2 ( 1) 2 4 (1 0 0) 2 2 5 x x e x dx x e x e e = + = + = 9. 2 1, 1 , 0 2 y x y x x = = ( ) ( ) ( ) 1 2 2 2 0 1 1 2 2 2 0 1 1 2 2 3 3 2 0 1 1 1 1 (1 ) 2 2 2 2 2 3 3 2 1 1 8 4 1 1 2 (0 0 0) 4 2 2 3 3 2 3 2 3 x x dx x x dx x x dx x x dx x x x x x x + = + + = + + = + + + = 11. 3 1, 1 , 2 2 y x y x x = = ( ) 1 2 3 3 2 1 1 2 2 4 4 2 2 1 1 1 1 (1 ) 2 2 2 4 4 2 1 1 1 1 2 ( 4 2 4) (4 4 2) 2 2 4 4 2 29 2 x x dx x x dx x x x x x x + = + + = − − + + + =
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Chapter 5 Applications of the Definite Integral 154 13. 2 2 1, 7 y x y x = = ( ) 2 2 2 2 2 3 2 7 1 2 8 3 16 16 16 16 3 3 64 3 x x dx x x ª º ¬ ¼ § · = ¨ ¸ © ¹ § · § · = + ¨ ¸ ¨ ¸ © ¹ © ¹ = ³ 15. 2 1, 3 1 y x y x = + = ( ) 2 2 1 2 2 3 1 3 1 1 3 2 2 3 12 8 3 1 4 2 2 3 2 3 1 6 x x dx x x x ª º + ¬ ¼ § · = ¨ ¸ © ¹ § · § · = ¨ ¸ ¨ ¸ © ¹ © ¹ = ³ 17. 3 , 3 2 y x y x = = + ( ) 2 3 1 2 2 4 1 3 2 3 2 2 4 3 1 (6 4 4) 2 2 4 27 4 x x dx x x x + § · = + ¨ ¸ © ¹ § · = + ¨ ¸ © ¹ = ³ 19. 3 2 , y x y x = = ( ) 1 3 4 1 2 3 0 0 3 4 1 1 (0 0) 3 4 1 12 x x x x dx § · = ¨ ¸ © ¹ § · = ¨ ¸ © ¹ = ³ 21. 2 , 1 x y e y x = = 0 2 .7145 0 3 .7145 [(1 ) ] 3 ( 1 0 0) ( 1.08235) .08235 x x x e dx x e x § · = + ¨ ¸ © ¹ = − + − − = ³ 23. 2 sin , y x y x = = ( ) .8767 3 .8767 2 0 0 sin cos 3 .135697 x x x dx x § · = ¨ ¸ © ¹ = ³ 25. 4 , 2 y x y x = = + ( ) 1.3532 2 5 1.3532 4 1 1 2 2 2 5 4.01449 x x x x dx x § · + = + ¨ ¸ © ¹ = ³ 27. , 2 , 0 2 y x y x y x y = = = = ( ) 1 1 0 0 1 2 0 [(2 ) ] (2 2 ) 2 (2 1) (0 0) 1 y y dy y dy y y = = = = ³ ³ 29. 2 3 , 2 x y x y = = + ( ) 2 2 1 2 3 2 1 3 2 3 2 2 3 8 3 1 6 4 2 3 2 3 1 6 y y dy y y y ª º + ¬ ¼ § · = ¨ ¸ ¨ ¸ © ¹ § · § · = ¨ ¸ ¨ ¸ © ¹ © ¹ = ³ 31. 1 1 2 0 0 , , 1 2 1 0 1 x y x y x xdx x = = − = = = = ³ 33. , 2, 6 , 0 y x y y x y = = = = ( ) 2 2 0 0 2 2 0 (6 ) (6 2 ) 6 (12 4) (0 0) 8 y y dy y dy y y = = = = ³ ³
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Chapter 5 Applications of the Definite Integral 155 ( ) ( ) ( ) ( ) 3 3 3 2 0 0 3 3 3 2 3 3 3 3 3 3 3 3 (0 0) 2 3 3 3 3 (9 9) 3 3 3 2 3 x x dx x x x dx x § · = ¨ ¸ © ¹ = = § · = ¨ ¸ © ¹ = = ³ ³ 37.
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