Question 20 and 32 please
es/117742/files/folder/Homeworks?preview=39002051 Carmen X 840b2c..: "Towar... SECTION 3.2 EXERCISES Review Questions 1. How does a line integral differ from the single-variable integral 17 . x2 + yds: C is the line segment from (1, I) to (10, 10). Jaf(x) dx? 2. How do you evaluate the line integral of ds, where C is param- cterized by a parameter other than arc length? 18 . (xy) 103 ds; C is the curve y = x2, for 0 5 x 3 1. 3. If a curve C is given by r(1) - (t, r?), what is |r'(1) |? 19. 4. xy ds; C is the portion of the ellipse * + 16 = 1 in the first Given a vector field F and a parameterized curve C, explain how to evaluate the line integral / F . T ds. quadrant, oriented counterclockwise. 5. . How can J F . T ds be written in the alternative form 20. (2 - 3y) ds; C is the line segment from (-1, 0) to (0, 1) So(fix ( t) + 8(ily' (1) + h(1)z (1)) an? 6. Given a vector field F and a closed smooth oriented curve C. followed by the line segment from (0, I ) to (1. 0). what is the meaning of the circulation of F on C? 21-24. Average values Find the average value of the following 7. How is the circulation of a vector field on a closed smooth functions on the given curves. oriented curve calculated? 21. f(x, y) = x + 2y on the line segment from (1, 1) to (2,5) 8. Given a two-dimensional vector field F and a smooth oriented 22. f(x, y) = x2 + dy on the circle of radius 9 centered at the origin curve C, what is the meaning of the flux of F across C? 23. f(x, y) = V4 + 9y2 on the curve y = x1/, for 0 5 x 5 5 9. How do you calculate the flux of a two-dimensional vector field across a smooth oriented curve C? 24. f(x, y) = xe' on the unit circle centered at the origin 10. Sketch the oriented quarter circle from (1, 0) to (0, 1) and supply 25-30. Scalar line integrals in R' Convert the line integral to an ordi- a parameterization for the curve. Draw the unit normal vector nary integral with respect to the parameter and evaluate it. at several points on the curve. 25. (x + y + z) ds; C is the circle r(() = (2 cost, 0, 2 sint), for Basic Skills OSIS 2T. 11-14. Scalar line integrals with arc length as parameter Evaluate the following line integrals. 26. (x - y + 2z) ds; C is the circle r() = (1, 3 cos /, 3 sin (), 11. xy ds; C is the unit circle r(s) = (coss, sins), for for 0 S I S 217. OSSS 2 T . 27. xyzds; C is the line segment from (0, 0, 0) to (1, 2, 3). 12. ( x + y) ds; C is the circle of radius I centered at (0, 0). 28. ds; C is the line segment from (1, 4, 1) to (3, 6,3). 13 . ( x2 - 2y?) ds; C is the line r(s) = (s/ V2,s/V2), for 29. (y - z) ds; C is the helix r(1) - (3 cost, 3 sin , (), for OSS54. OSIS 2 T. 14 . x y ds ; C is the line r (s ) = (s/ V2, 1 - s/ V2), for 30. xe" ds; Cisr(1) - ( 1, 21 , - 41 ) , for I s . 5 2. OSSS 4. 15-20. Scalar line integrals in the plane 31-32. Length of curves Use a scalar line integral to find the length of the following curves. a. Find a parametric description for C in the form r(1) = (x(1), y(1) ). 31. r(1) = 20 sin . 20 cos 41 2) 15). for 0 5 1 = 2 b. Evaluate [r' (1) . c. Convert the line integral to an ordinary integral with respect to the 32. r(1) = (30 sin t, 40 sin (, 50 cost), for 0 5 1 5 277 parameter and evaluate it. 33-38. Line integrals of vector fields in the plane Given the follow- 15. ( x2 + y? ) ds; C is the circle of radius 4 centered at (0, 0). ing vector fields and oriented curves C. evaluate J F . T ds. 33. F = (x, y) on the parabola r(1) (4t. (?), for 0 SIS 1 16. (x2 + 12 ) ds ; C is the line segment from (0, 0) to ( 5, 5 ) . 34. F = (-y,x) on the semicircle r(1) = (4 cost, 4 sint), for OSIST
