Fall 2007 - Hohnhold's Class - Exam 1 (Version A)

# Fall 2007 - Hohnhold's Class - Exam 1 (Version A) - Midterm...

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Midterm 1 for Math 20B Fall Quarter 2007, UCSD Henning Hohnhold Name: Section: #1: #2: #3: #4: #5: Total: (1) Computing integrals. (10 points) Please evaluate the following integrals: ( a ) Z 6 · ln( x ) · ( x + 1) dx and ( b ) Z cos( e x ) · e x · sin( e x ) dx. (a) We use integration by parts to evaluate the integral: Z 6 · ( x + 1) · ln( x ) dx = 6 2 ( x + 1) 2 · ln( x ) - Z 6 2 ( x + 1) 2 · 1 x dx = 3( x + 1) 2 · ln( x ) - 3 Z x + 2 + 1 x dx = 3( x + 1) 2 · ln( x ) - 3 ± 1 2 x 2 + 2 x + ln | x | ² + C (b) We substitute u = sin( e x ). Then du = e x cos( e x ) and Z cos( e x ) · e x · sin( e x ) dx = 1 2 u 2 + C = 1 2 sin 2 ( e x ) + C It is also possible to make two simpler substitutions to get the same result. A diﬀerent substitution gives - 1 2 · cos 2 ( e x ) + C . This is okay, since sin 2 + cos 2 = 1, i.e. the functions - 1 2 · cos 2 ( e x ) and 1 2 · sin e x only diﬀer by a constant.

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(2) Using integrals to compute areas. (10 points) Please sketch the region en- closed by the curves x = - sin( πy ) (for - 1 2 y 1) , y = 1 x + 1 , and x = 1 and compute its area. Hint: x = - sin ( πy ) and y = 1 x +1 intersect in the point (0 , 1) . We integrate in y -1 -0.5 0 0.5 1 -0.5 0.5 1 direction and split the area into two parts in order to compute it: the part below
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## This note was uploaded on 04/29/2008 for the course MATH 20B taught by Professor Justin during the Fall '08 term at UCSD.

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Fall 2007 - Hohnhold's Class - Exam 1 (Version A) - Midterm...

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