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

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

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Unformatted text preview: 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 3 · ( x- 1) · ln( x ) dx and ( b ) Z cos( x 1 2 ) · x- 1 2 · sin( x 1 2 ) dx. (a) We use integration by parts to evaluate the integral: Z 3 · ( x- 1) · ln( x ) dx = 3 2 ( x- 1) 2 · ln( x )- Z 3 2 ( x- 1) 2 · 1 x dx = 3 2 ( x- 1) 2 · ln( x )- 3 2 Z x- 2 + 1 x dx = 3 2 ( x- 1) 2 · ln( x )- 3 2 1 2 x 2- 2 x + ln | x | + C (b) We substitute u = sin( x 1 2 ). Then du = 1 2 x- 1 2 cos( x 1 2 ) and Z cos( x 1 2 ) · x- 1 2 · sin( x 1 2 ) dx = 2 1 2 u 2 + C = sin 2 ( x 1 2 ) + C It is also possible to make two simpler substitutions to get the same result. A different substitution gives- cos 2 ( x 1 2 ) + C . This is okay, since sin 2 +cos 2 = 1, i.e. the functions- cos 2 ( x 1 2 ) and sin 2 ( x 1 2 ) only differ by a constant. (2) Using integrals to compute areas. (10 points) Please sketch the region en- closed by the curves x =- cos( πy 2 ) (for- 2 ≤ y ≤ 1) , y = 1 x + 1 , and x = 1 and compute its area. Hint: x =- cos( πy 2 ) and y = 1 x +1 intersect in the point (0 , 1) ....
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Fall 2007 - Hohnhold's Class - Exam 1 (Version B) - Midterm...

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