APPM 1360 Midterm: appm1360spring2018exam4sol
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1. (a) (12 pt) solve the differential equation e x/2 dy dx (b) (10 pt) evaluatez 1/2 (1 x2)3/2 dx. Solution: (a) x y2 for the function y. x y2 e x/2 dy dx. 3 x ex/2 dx dv=ex/2 dx du=dx v=2ex/2. = 2xex/2 4ex/2 + c y = 3q3(cid:0)2xex/2 4ex/2 + c(cid:1) (b) let x = sin , dx = cos d . 0 cos (cid:0)1 sin2 (cid:1)3/2 sec2 d = [tan ] /6 cos cos3 d . 0 : (12 pt) consider the region enclosed by the curve x2. + y2 = 1. (a) sketch the region. Label all intercepts. (b) suppose the region is rotated about the line x = 4. Use the shell method to set up an integral to. 9 ! dx: (18 pt) determine whether the series is convergent or divergent. (a) The function 1/(cid:0)x(ln x)2(cid:1) is positive, continuous and decreasing on. 100 dx x(ln x)2 = lim t z t.
