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2.3 Double Integrals in Polar Coordinates 137 SECTION 2.3 EXERCISES 15. R - { (r,0): 0 = = 2,0 0 5 27 ) Review Questions 1. Draw the polar region ( (, ): 1 = r s 2.0 - 8 - #/2). Why is it called a polar rectangle? 2. Write the double integral Jef(x. >) A as an iterated integral in polar coordinates when R - { (re): a s r s b, a s es s). Sketch in the ay-plane the region of integration for the integral Explain why the clement of area in Cartesian coordinates dx dy becomes r dr de in polar coordinates, How do you find the area of a polar region 16. R - { (r.0) :0 s / = 1,0 = =T) How do you find the average value of a function over a region 17. R - [(. 0): V3 = = 2 V2, 0 = 0 = 27) that is expressed in polar coordinates? 18. R = ((r.8): V3 S / S V15. -7/2 5 8 = 7) Basic Skills 19-22. Volume between surfaces Find the volume of the following solids. 7-10. Polar rectangles Sketch the following polar rectangles. 19. The solid bounded by the paraboloids z - x + y and 7. R - ((re):0 s r s 5.0 5 0 5 /2) z - 2 - 6-32 R = [(.0): 2 5 / S 3, /4 5 0 5 57/4) 9. R- ((8): 1 5 5 4, -W/4 5 8 5 2m/3) 10. R = [(r.0): 4 5 / $ 5, 7/3 50s T/2} z - 2-12-32 11-14. Solids bounded by paraboloids Find the volume of the solid below the paraboloid : - 4 - x - y and above the following polar rectangles. 11. R = {(7.0):0 5 r s 1,0 5 8 5 2) 20. The solid bounded by the paraboloids z - 2x2 + y? and z - 27 - .x2- 2y2 2 = 27 - x2- 2x 12. R = [ (r.e):0 s r s 2,0 s 8 s 2m} 13. R = { (r.0): 1 s r s 2,0 s 0 s 27} 14. R = [(r, 8): 1 S rs 2, -m/2 5 0 s #/2} 20 15-18. Solids bounded by hyperboloids Find the volume of the solid - 2+2 + 32 below the hyperboloid z = 5 - V1 + x3 + y and above the follow- ing polar rectangles. 20 20 =39002037