MATH 241 Study Guide - Midterm Guide: Divergence Theorem, Iterated Integral, Spherical Coordinate System
10 Jan 2019
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Solutions and grading key: (30 points) find the area in the rst quadrant bounded on the left by the line y = x and above and below by the curves yx2 = 1 and yx2 = 2. 4 (the region extends out to in nity, but its area is nite. ) The change of variables u = x/y, v = yx2 may help (though you can do the problem other ways also). The region becomes u 1 and 1 v 2. (doing things this way, 6 for the correct limits, 14 for the jacobian, 10 for the evaluation. ) Now one can compute the matrix of partial derivatives: (cid:18)xu xv yu yv(cid:19) =(cid:18) 1. 9 u 4/3v 1/3 = 1 which has determinant 1 since this is positive. (method 2) find the matrix of partial derivatives: There is no need to change the sign (cid:18)ux uy vy(cid:19) =(cid:18)1/y x/y2 x2 (cid:19) , 2xy vx which has determinant x2/y + 2x2/y = 3x2/y.
