227_solns_to_sample_final[1]

# 227_solns_to_sample_final[1] - Math 227.04 Solutions to...

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Unformatted text preview: Math 227.04 Solutions to Sample Final Exam 1. (10 pts each) Compute each of the following integrals. Show all steps (a) Z x 3 p 1 + x 2 dx = 1 2 R x 2 & 1 + x 2 ¡ 2 xdx = 1 2 R ( u & 1) u 1 = 2 du (where u = 1 + x 2 ; du = 2 xdx , and x 2 = u & 1 : ) : Thus Z x 3 p 1 + x 2 dx = 1 2 Z ( u & 1) u 1 = 2 du = 1 2 Z ¢ u 3 = 2 & u 1 = 2 £ du = 1 2 ¤ 2 5 u 5 = 2 & 2 3 u 3 = 2 ¥ + C = 1 5 & 1 + x 2 ¡ 5 = 2 & 1 3 & 1 + x 2 ¡ 3 = 2 + C (b) Assign parts u = ln x; du = 1 x dx ; dv = xdx; v = 1 2 x 2 : Then 2 Z 1 x ln xdx = 1 2 x 2 ln x j 2 1 & Z 2 1 1 2 xdx = (2 ln 2 & 0) & ( 1 4 x 2 j 2 1 ) = 2 ln 2 & 1 4 2 2 + 1 4 = 2 ln 2 & 3 4 : (c) Z x 2 ( x 2 & 1) dx Apply partial fractions. x 2 ( x 2 & 1) = 1 + 1 ( x & 1)( x +1) = 1 + 1 = 2 ( x & 1) & 1 = 2 x +1 : Thus Z x 2 ( x 2 & 1) dx = Z (1 + 1 = 2 ( x & 1) & 1 = 2 x + 1 ) dx = x + 1 2 ln j x & 1 j & 1 2 ln j x + 1 j + C: (d) Let u = & x 2 , du = & 2 xdx: Thus Z b xe & x 2 dx = & 1 2 Z b e & x 2 ( & 2 dx ) = & 1 2 Z & b 2 e u du 1 2 ¢ 1 & e & b 2 £ 1 Z xe & x 2 dx = lim b !1 b Z xe & x 2 dx = lim b !1 1 2 ¢ 1 & e & b 2 £ = 1 2 : 1 2. (10 pts each) Let R denote the portion of the xy & plane on or above the line y = 1 and below the semicircle y = p 4 & x 2 : Suppose R is revolved about the y axis. Set up the integral for the resulting volume using each of the following methods. In both cases, include sketches to illustrate your cross-volume using each of the following methods....
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227_solns_to_sample_final[1] - Math 227.04 Solutions to...

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