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

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

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

View Full Document Right Arrow Icon
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.
Background image of page 1

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

View Full DocumentRight Arrow Icon
(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
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 04/29/2008 for the course MATH 20B taught by Professor Justin during the Fall '08 term at UCSD.

Page1 / 6

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