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Unformatted text preview: B U Department of Mathematics Math 102 Calculus II Spring 2006 First 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 area of the region inside the circle r = 3 cos θ and outside the cardioid r = 1 + cos θ . Solution:- 6 ? r = 3 cos θ r = 1 + cos θ @ @ I • 3 • 2 • 1 • 1 •- 1 θ = π/ 2 θ = 0 The first step is to determine the intersections of the cardioid r = 1 + cos θ and the circle r = 3 cos θ , since this information is needed for the limits of integration. To find points of intersection, we can equate the two expressions for r . This yields, 3 cos θ = 1 + cos θ ⇒ cos θ = 1 2 which is satisfied by the positive angles θ = π 3 and θ = 5 π 3 ....
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