491 SECTION 7.3 Trigonometric Substitution We now substitute u − 2 sin , giving du − 2 cos d and s4 2 u 2 − 2 cos , so Figure 5 shows the graphs of the integrand in Example 7 and its indefinite integral (with C − 0). Which is which? y 3 x 2 sin 2 1 dx − y 2 cos d 2 2 cos s3 2 2x 2 x − 2 _4 y s2 sin 2 1d d − 22 cos 2 1 C − 2s4 2 u 2 2 sin21 _5 SD u 2 FIGURE 5 − 2s3 2 2x 2 x 2 2 sin21 1–3 Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. 1. 2. 3. dx y x − 2 sin x s4 2 x 2 2 x3 y sx 2 1 4 dx x − 2 tan x − 2 sec 4. 5. 7. y s9 2 x 2 y y 3 y 1y2 0 9. 2 11. 0 13. 15. a 0 y a.0 dx sx 2 2 1d3y2 x 3 0 s36 2 x 2 21. y 8. y 10. y 2y3 y 2 dt x s1 2 4x dx 2 0 s4 1 t 2 y 1 x 2 sa 2 2 x 2 dx 12. 14. y x11 2 18. y s1 1 x 2 dx x 20. y 22. y 1 y 1 x2 0.6 s9 2 25x 2 dx dx 24. sx 1 2x 1 5 25. yx 27. y sx 29. y x s1 2 x 2 s1 1 x 2 0 s3 1 2x 2 x 2 dx 2 x 0 2 1 2x dx Q dx fsaxd2 2 b 2 g 3y2 dx sx 2 7 2 1C dx sx 2 1 1 dx sx 2 x 2 dx x2 26. y s3 1 4x 2 4x 28. y sx 30. y 2 2 3y2 d dx x2 1 1 dx 2 2x 1 2d2 4 dx cos t y2 0 s1 1 sin 2 t dt t 2st 2 2 16 0 16. y S D dx dt 0 sx 2 2 9 dx x3 y y 6. dx , sa 2 1 x 2 d3y2 a y dx sx 2 2 1 dx x4 y 19. 23. 4–30 Evaluate the integral. x2 x y 0 sx 2 2 4 dx x y 17. 1C s4 2 9x 2 dx dx sx 2 1 1d2 2y3 s2y3 dx x 5s9x 2 2 1 31. (a) Use trigonometric substitution to show that y dx sx 1 a 2 2 − ln ( x 1 sx 2 1 a 2 ) 1 C (b) Use the hyperbolic substitution x − a sinh t to show that y dx sx 1 a 2 2 SD − sinh21 x a 1C These formulas are connected by Formula 3.11.3. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 492 CHAPTER 7 Techniques of Integration (b) Use the figure to give trigonometric interpretations of both terms on the right side of the equation in part (a). 32. Evaluate x2 dx sx 1 a 2 d3y2 y 2 y a a@-t@ y=œ„„„„„ (a) by trigonometric substitution. (b) by the hyperbolic substitution x − a sinh t. ¨ ¨ 33. Find the average value of f sxd − sx 2 1yx, 1 < x < 7. 2 34. Find the area of the region bounded by the hyperbola 9x 2 2 4y 2 − 36 and the line x − 3. 35. Prove the formula A − 12 r 2 for the area of a sector of a circle with radius r and central angle . [Hint: Assume 0 , , y2 and place the center of the circle at the origin so it has the equation x 2 1 y 2 − r 2. Then A is the sum of the area of the triangle POQ and the area of the region PQR in the figure.] y 0 x 40. The parabola y − 12 x 2 divides the disk x 2 1 y 2 < 8 into two parts. Find the areas of both parts. 41. A torus is generated by rotating the circle x 2 1 s y 2 Rd2 − r 2 about the x-axis. Find the volume enclosed by the torus. 42. A charged rod of length L produces an electric field at point Psa, bd given by EsPd − y L2a 2a P t b dx 4«0 sx 2 1 b 2 d3y2 where is the charge density per unit length on the rod and «0 is the free space permittivity (see the figure). Evaluate the integral to determine an expression for the electric field EsPd. ¨ O Q R x y P (a, b) ; 36. Evaluate the integral y 0 L x dx x sx 2 2 2 4 Graph the integrand and its indefinite integral on the same screen and check that your answer is reasonable. 37. Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves y − 9ysx 2 1 9d, y − 0, x − 0, and x − 3. 43. Find the area of the crescent-shaped region (called a lune) bounded by arcs of circles with radii r and R. (See the figure.) r R 38. Find the volume of the solid obtained by rotating about the line x − 1 the region under the curve y − x s1 2 x 2 , 0 < x < 1. 39. (a) Use trigonometric substitution to verify that y x 0 sa 2 2 t 2 dt − 21 a 2 sin21sxyad 1 12 x sa 2 2 x 2 44. A water storage tank has the shape of a cylinder with diameter 10 ft. It is mounted so that the circular cross-sections are vertical. If the depth of the water is 7 ft, what percentage of the total capacity is being used? Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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