# M19BSpr14FinKey.pdf - Math 19B Spring 2014 FINAL EXAM KEY...

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Math 19B, Spring 2014 FINAL EXAM KEY UCSC, Frank B¨ auerle, 6-12-14 1. Evaluating the integrals. (b)
(c) 4 Z 0 tan 3 ( ) sec 2 ( ) d = 4 Z 0 (tan | {z } u ) 3 du z }| { sec 2 ( ) d = 1 Z 0 u 3 du = u 4 4 1 0 = 1 4 2. For this problem let R be the region between the x -axis and the graph of f ( x ) = sin( x 2 ) where 0 x p .
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3. 1 Z 3 1 x 2 dx = lim b !1 b Z 3 1 x 2 dx = lim b !1 - 1 x b 3 = lim b !1 1 3 - 1 b = 1 3 4. The suitable trig substitution to evaluate Z x 2 p 4 - x 2 dx is x = 2 sin t , which gives dx = 2 cos t dt and the integral becomes Z x 2 p 4 - x 2 dx = Z 4 sin 2 t p 4 - 4 sin 2 t | {z } =2 cos