This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Valenica, Daniel Homework 9 Due: Oct 30 2007, 3:00 am Inst: Cheng 1 This printout should have 15 questions. Multiplechoice questions may continue on the next column or page find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Evaluate the iterated integral I = Z 4 1 n Z 4 1 ( x + y ) 2 dx o dy . 1. I = ln 5 2 correct 2. I = 2 ln 12 7 3. I = 2 ln 5 2 4. I = 1 2 ln 5 2 5. I = ln 12 7 6. I = 1 2 ln 12 7 Explanation: Integrating the inner integral with respect to x keeping y fixed, we see that Z 4 1 ( x + y ) 2 dx = h 1 x + y i 4 = n 1 y 1 4 + y o . In this case I = Z 4 1 n 1 y 1 4 + y o dy = h ln y ln(4 + y ) i 4 1 . Consequently, I = ln (4)(1 + 4) (4 + 4) = ln 5 2 . keywords: iterated integral, rational function, log integral 002 (part 1 of 1) 10 points Evaluate the iterated integral I = Z ln 5 Z ln 4 e 2 x y dx ! dy . 1. I = 4 2. I = 5 3. I = 6 correct 4. I = 7 5. I = 8 Explanation: Integrating with respect to x with y fixed, we see that Z ln 4 e 2 x y dx = 1 2 h e 2 x y i ln 4 = 1 2 e 2 ln 4 y e y = 4 2 1 2 e y . Thus I = 15 2 Z ln 5 e y dy = 15 2 h e y i ln 5 = 15 2 e ln 5 1 . Consequently, I = 15 2 1 5 1 = 6 . keywords: 003 (part 1 of 1) 10 points Valenica, Daniel Homework 9 Due: Oct 30 2007, 3:00 am Inst: Cheng 2 Determine the value of the double integral I = Z Z A 3 xy 2 9 + x 2 dA over the rectangle A = n ( x, y ) : 0 x 2 , 4 y 4 o , integrating first with respect to y . 1. I = 32 ln 13 9 2. I = 64 ln 13 18 3. I = 64 ln 9 13 4. I = 64 ln 13 9 correct 5. I = 32 ln 13 18 6. I = 32 ln 9 13 Explanation: The double integral over the rectangle A can be represented as the iterated integral I = Z 2 Z 4 4 3 xy 2 9 + x 2 dy dx , integrating first with respect to y . Now after integration with respect to y with x fixed, we see that Z 4 4 3 xy 2 9 + x 2 dy = h xy 3 9 + x 2 i 4 4 = 128 x 9 + x 2 . But Z 2 128 x 9 + x 2 dx = h 64 ln(9 + x 2 ) i 2 . Consequently, I = 64 ln 13 9 . keywords: 004 (part 1 of 1) 10 points Evaluate the integral, I , of the function f ( x, y ) = 2 xe xy over the rectangle A = { ( x, y ) : 0 x 3 , y 3 } . 1. I = 1 3 e 9 8 2. I = 1 3 e 9 10 3. I = 1 3 e 9 9 4. I = 2 3 e 9 8 5. I = 2 3 e 9 10 correct 6. I = 2 3 e 9 9 Explanation: The integral is given by I = Z Z A 2 xe xy dxdy. Since the integral with respect to y can be evaluated easily using substitution (or di rectly making the substitution in ones head), while the integral with respect to x requires integration by parts, this suggests that we should represent the double integral as the repeated integral I = Z...
View
Full
Document
This homework help was uploaded on 04/16/2008 for the course CALC 303L taught by Professor Cheng during the Fall '07 term at University of Texas at Austin.
 Fall '07
 CHENG

Click to edit the document details