MTH 162 Final: MTH 162 University of Rochester pr Final Solutions Fall 05

• School: University of Rochester
• Department: Mathematics
• Course: MTH 162
• Professor: All

Document Summary

Part a: (30 points) (a) (10 points) calculate (b) (10 points) calculate (c) (10 points) calculate. Answer: (a) use the substitution y = x2, dy = 2xdx to get. 2 cos(y) + c cos(x2) + c (b) we could use the same substitution as in part (a). On the other hand, ln(x2) = 2 ln(x). So, using integration by parts with u = ln(x) dv = x dx du = Z x ln(x2) dx = 2z x ln(x) dx x2. 1 x dx (c) we use partial fractions. Setting y = 0, we get a = 1; setting y = 1 we get b = 1/2; setting y = 1 we get c = 1/2. The formula for the surface area of curves rotated about lines parallel to the y-axis gives a. 2 rpdx2 + dy2 (x ( 1))s1 +(cid:18) dy dx(cid:19)2 (x + 1)p1 + x2(x2 + 2) dx (x + 1) x4 + 2x2 + 1 dx.