MATH 141 Midterm: MATH141 ROSENBERG-J FALL2007 0101 MID SOL

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Wednesday, september 19, 2007: (25 points) find the volume of the solid whose base is the region in the x-y plane where. 4x2 + y2 1, such that each cross-section perpendicular to the x-axis is a solid square. For xed x, the cross-section has height 2 1 4x2 and thus cross-sectional area a(x) = (cid:2)2 1 4x2(cid:3)2. The range of values of x is given by the inequality 4x2 1, or x2 1. 2 (1 4x2) dx = 8hx . As one can see, the region is symmetric around the x-axis, so y = 0 (without our needing to do any calculation). Now we observe that the region projects to the region 0 x 4 and is bounded above and below by y = 4 x. 3(cid:2)u3/2(cid:3)4 (here we"ve used the change of variables u = 4 x, du = dx. ) Similarly, (4 u) u du = 2z 4. 0 (cid:0)4u1/2 u3/2(cid:1) du u5/2i4 u3/2 .

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