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

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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) ....
View Full Document

{[ snackBarMessage ]}

Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online