FinalExamReviewSpring09

# - CALCULUS 4 Final Exam REVIEW Part 1 SPRING 09(1 Determine the following about double integrals in polar coordinates(a Write a double integral in

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CALCULUS 4 Final Exam REVIEW Part 1 / SPRING 09 (1.) Determine the following about double integrals in polar coordinates. (a.) Write a double integral in Polar Coordinates that equals the area of the region above the line: y1 = and inside the circle : x 2 y 2 + 4 = . 0 30 60 90 120 150 180 210 240 270 300 330 4 2 0 Polar Curves x r cos θ () = y r sin θ = = r sin θ = r1 θ 1 sin θ = csc θ = x 2 y 2 + 4 = r2 θ 2 = Page 1 of 12

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r1 θ () r2 θ = 1 sin θ 2 = sin θ 1 2 = θ 1 π 6 = θ 2 5 π 6 = A π 6 5 π 6 θ csc θ 2 r r d d = (b.) Write a double integral in Polar Coordinates that equals the area of the region enclosed by the rose : r θ sin 2 θ = . 0 45 90 135 180 225 270 315 2 1 0 r θ θ Page 2 of 12
A4 0 π 2 θ 0 sin 2 θ () r r d d = (c.) Write a double integral in Polar Coordinates that equals the area of the region inside the circle: r1 θ 1 = and outside the cardioid: r2 θ 1 cos θ + = . 0 45 90 135 180 225 270 315 3 2 1 0 Polar Region r1 θ r2 θ θ A π 2 3 π 2 θ 1 cos θ + 1 r r d d = Page 3 of 12

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(d.) Write a double integral in Polar Coordinates that equals the area of the region common to all three circles: r1 θ () 1 = , r2 θ 2 sin θ = and r3 θ 2 cos θ = . A c 0 π 6 θ 0 2sin θ r r d d π 6 π 3 θ 0 1 r r d d + π 3 π 2 θ 0 2 cos θ r r d d + = 0 45 90 135 180 225 270 315 3 2 1 0 Polar Areas r1 θ r2 θ r3 θ θ (e.) Write a double integral in Polar Coordinates that equals the volume of the region inside x 2 y 2 + z 2 + 9 = and outside x 2 y 2 + 1 = Page 4 of 12
3 2 10123 3 2 1 1 2 3 Annular Region of Integration x y zr θ , () 9r 2 = V2 0 2 π θ 1 3 r 2 r d d = Page 5 of 12

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x y z Volume between Sphere & Cylinder (f.) Write a double integral in Polar Coordinates that equals the volume of the region inside x 2 y 2 + 2y = , below zx 2 y 2 + = , and above z0 = . r θ () 2 sin θ = Page 6 of 12
20 2 2 2 v 0 π θ 0 2 sin θ () r rr d d = x y z Volume of Cylinder below Cone Page 7 of 12

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(g.) Write a double integral in Polar Coordinates that equals the volume of the region inside x 2 y 2 + x 0 = , below z1x 2 y 2 = , and above z0 = . 1 0.5 0 0.5 1 1 0.5 0.5 1 Region of Polar Integration x 2 y 2 + x = r 2 r cos θ () = r θ cos θ = I π 2 π 2 θ 0 cos θ r 1r 2 r d d = Page 8 of 12
x y z Volume of Cylinder below Surface (h.) Use Polar Coordinates to evaluate y x 9x 2 y 2 d d over the region "R", where "R" is in the first quadrant within the circle: x 2 y 2 + 9 = . I 0 π 2 θ 0 3 r 9r 2 r d d = Page 9 of 12

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Let : u9r 2 = , then : du 2 rdr = I 1 2 0 π 2 θ 9 0 u u d d = 1 2 0 π 2 θ 0 9 u u 1 2 d d = 0 9 u u 1 2 d 2 3 27 () = 18 = I9 0 π 2 θ 1 d = 9 2 π = (i.) Convert the integral 0 2 y y 4y 2 x 1 1x 2 + y 2 + d d to Polar Coordinates.
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## This note was uploaded on 07/06/2011 for the course MATH 200 taught by Professor Kingsberry during the Spring '08 term at Drexel.

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- CALCULUS 4 Final Exam REVIEW Part 1 SPRING 09(1 Determine the following about double integrals in polar coordinates(a Write a double integral in

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