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Unformatted text preview: ) and z = 1 + x 2 + y 2 . 1 4. Find a parametrization for the surface deﬁned by the intersection of the plane x + y + z = 1 with the cylinder x 2 + y 2 = 1. Use that parametrization to calculate the area of the surface. 5. Suppose that a particle follows the path r ( t ) = 2 cos(2 t ) i + 2 sin(2 t ) j + 3 t k . Then ﬁnd the total length of the path travelled by the particle from t = 0 to t = π/ 4. 6. Set up a double integral in polar coordinates to ﬁnd the volume of the solid which is bounded below by the paraboloid z = x 2 + y 2 and above by the plane z = 2 y . DO NOT EVALUATE! 7. Evaluate the integral Z 1 Z 1 y cos ± 1 2 πx 2 ² d x d y. 8. Find the volume of the region under the surface z = x 2 + 2 y and over the region in the ﬁrst quadrant between the line segment connecting (0 , 2) and (2 , 0) and the curve y = 4x 2 . 2...
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 Spring '09
 Integrals, Polar coordinate system, Prof. Qiao Zhang, double integral x2

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