Applications of Definite Integral

Applications of Definite Integral - Contents 12...

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Unformatted text preview: Contents 12 Applications of the Definite Integral 179 12.1 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 12.2 Solids of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 12.3 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 12.4 Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 12.5 Liquid Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 12.6 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 2 CONTENTS Chapter 12 Applications of the Definite Integral 12.1 Area In problems 1 to 15 find the area of the region under the following curves between the given values of x . Use the principles of graph sketching taught in your course to sketch the required region. Label your diagram. 1. y = 5- 1 2 x ; x =- 6 , x = 0 2. y = 1 2 x 2 ; x = 0 , x = 3 3. y = x 2 + 3; x =- 2 , x = 2 4. y = 10 x- x 2 ; x = 1 , x = 10 5. y = √ x ; x = 0 , x = 4 6. y = 2 √ x- 1; x = 1 , x = 10 7. y = x ( x- 5) 2 ; x = 0 , x = 2 8. y = ( x + 1) 3 + 1; x =- 2 , x = 0 9. y = 2 x + 1 x 2 ; x = 1 , x = 3 10. y = 5 √ x +2 ; x = 0 , x = 5 11. y = x √ 4 x 2 + 1; x = 0 , x = 2 12. y = x c √ c 2- x 2 ; x = c, where c > 13. y = ( ( x + 1) 3 ,- 1 ≤ x ≤ , 1- x 2 4 , < x ≤ 2; x =- 1 , x = 2 14. y = ( 1 2 ( x + 3) ,- 3 ≤ x < 1 , 3- x, 1 ≤ x ≤ 3; x =- 3 , x = 3 15. y = | x | ; x =- 2 , x = 5 In problems 16 to 64 find the area of the region or regions bounded by the following curves. Use the principles of graph sketching taught in your course to sketch the required region(s). Label your diagram. 16. y = x 3 , y = 0 , x = 1 , x = 2 17. y = x 2 + 4 x, y = 0 18. y = x 2- 4 x + 2 , x + y- 6 = 0 19. y =- x 2- 6 x, y = 2 x 20. y = x 2- 6 , y = 6- z 2 21. y = x 3 , y = 1 , x = 0 , x = 3 180 Applications of the Definite Integral 22. y = x ( x- 2) 2 , y = 0 23. y = x ( x + 3) 2 , y = 0 , x =- 3 , x =- 1 24. y = √ 4 + x, x = 0 , y = 0 25. y = x 3- x 2- 6 x, y = 0 26. x 3- 2 x 2- 11 x + 12 , y = 0 27. y = 9- x 2 , x- y + 7 = 0 28. y = 2 √ x, y =- 2 x + 12 , y = 0 29. y = 2 √ x, y =- 2 x + 12 , x = 0 30. y =- x 3 + 2 x 2 + 3 x, y =- 5 x 31. x 2 y = 3 , 4 x + 3 y- 13 = 0 32. y 2- 4 y- 2 x = 0 , x = 0 33. y 2 = x + 1 , y 2 = 7- x 34. y 2 = x + 4 , x- 2 y + 1 = 0 35. x = 4 y- y 2 , x + y = 4 36. x = y ( y- 3) 2 , x = 0 37. x = y 3- 4 y 2- y + 4 , x = 0 38. x = y 2- y 3 , x = 0 39. y 2 = x 3 , x = 0 , x = 1 40. y 2 = ax, x 2 = by , where a > 0 and b > 41. y = x 4- 3 x 2 , y = x 2 42. y 2 = x 2 ( x 2- 1) , x = 3 43....
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Applications of Definite Integral - Contents 12...

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