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Homework 11

# Homework 11 - x 2 − y 2 = 4 and the ellipses x 2 4 y 2 =...

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APPM 2350 HOMEWORK 11 FALL 2010 Due Friday, November 12, 2010 at 4 pm under your TA’s door. 1. Find the area enclosed by the ellipse E given by the equation x 2 a 2 + y 2 b 2 = 1 in the following way: (a) First, write the double integral that gives the area of E in Cartesian coordinates. Don’t evaluate this integral! (b) Next, transform the variables using x = au and y = bv . Clearly sketch the domain of E in the uv -plane. Rewrite the xy -integral from part (a) as an integral in the uv -plane. (c) Evaluate the new integral from part (b) (feel free to change coordinate systems) and show that the area of E is equal to π ab . (d) Extend the above techniques to find the volume enclosed by the ellipsoid E given by the equation x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 . 2. Consider the problem of evaluating the double integral I = D xy y 2 x 2 dA , where D is the region in the first quadrant bounded by the hyperbolas x 2 y 2 = 1,
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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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