mid2-S00-sol - My NAME is: Problem 1 2 3 4 Total Score MAT...

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Unformatted text preview: My NAME is: Problem 1 2 3 4 Total Score MAT 126 Calculus B Midterm 2, Solutions April 5, 2000 Show all your work on these pages! Total score = 100 1. (30 points) Evaluate each of the following definite integrals. (a) Z 1 sin( x ) dx Solution: Let u = x . Then du = dx . When x = 0 ,u = 0 , and when x = 1 , u = . Thus Z 1 sin( x ) dx = Z sin( u ) du =- cos( u ) =- cos( ) -- cos(0) = 1 + 1 = 2 (b) Z 1 x 1- x 2 dx Solution: Let u = 1- x 2 . Then du =- 2 xdx . When x = 0 , u = 1 and when x = 1 , u = 0 . Thus, Z 1 x 1- x 2 dx =- 1 2 Z 1 udu =- 1 2 2 3 x 3 2 1 =- 1 3- 1 ! = 1 3 . (c) Z 1- 1 xe x dx Solution: We integrate by parts. Take u = x and dv = e x dx . Then du = dx and v = e x . So we have Z 1- 1 xe x dx = xe x 1- Z 1- 1 e x dx = xe x- e x 1- 1 = ( e 1- e 1 )- (- e- 1- e- 1 ) = 2 e- 1 = 2 e . 1 2. (30 points) Find the following antiderivatives....
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mid2-S00-sol - My NAME is: Problem 1 2 3 4 Total Score MAT...

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