102y05mt2 - B U Department of Mathematics Math 102 Calculus...

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Unformatted text preview: B U Department of Mathematics Math 102 Calculus II Summer 2005 Second Midterm This archive is a property of Bo˘gazi¸ci University Mathematics Department. The purpose of this archive is to organise and centralise the distribution of the exam questions and their solutions. This archive is a non-profit service and it must remain so. Do not let anyone sell and do not buy this archive, or any portion of it. Reproduction or distribution of this archive, or any portion of it, without non-profit purpose may result in severe civil and criminal penalties. 1.) Find the volume of the part of the sphere of radius 3 that is left after drilling a cylindrical hole of radius 2 through the center. Solution: The sphere has equation: x 2 + y 2 = 9 In polar coordinates, r 2 + z 2 = 9 ⇒ z = ± √ 9 − r 2 V = Z 2 π Z 3 2 [ √ 9 − r 2 − ( − √ 9 − r 2 )] rdrdθ Let u = 9 − r 2 ⇒ du = − 2 rdr ⇒ V = − 1 2 Z 2 π Z 5 2 u 1 / 2 dudθ = − 2 3 Z 2 π − 5 3 / 2 dθ = 20 √ 5 π 3 2.)2....
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102y05mt2 - B U Department of Mathematics Math 102 Calculus...

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