Unformatted text preview: x 2 − y 2 = 4, and the ellipses x 2 / 4 + y 2 = 1, x 2 / 16 + y 2 / 4 = 1. (a) Sketch the region D in the xy-plane. (b) Select a variable transform u = u ( x, y ) and v = v ( x, y ), and sketch the region D in the uv-palne. (c) Rewrite the original integral in terms of your new varibles, u and v . (d) Evaluate your integral from part (c) to ±nd the value of I . 3. A metal wire lying in the xy-plane is bent in the shape of the semicircle x 2 + y 2 = 4, for y ≥ 0. The mass density (mass per unit length) at each point ( x, y, z ) of the wire is δ ( x, y, z ) = 3 − y . (a) Find the total mass of the wire, m T . (b) Find the x and y coordinates of the center of mass of the wire, (¯ x, ¯ y ). (c) Find the radius of gyration, R z , for the wire about the z-axis....
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- Fall '07
- Coordinate system, Polar coordinate system, integral xy