HW 9 Solution

# HW 9 Solution - Kim Jin – Homework 9 – Due 3:00 am –...

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Unformatted text preview: Kim, Jin – Homework 9 – Due: Oct 30 2007, 3:00 am – Inst: Diane Radin 1 This print-out should have 15 questions. Multiple-choice 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 2 ( x + y ) 2 dx o dy . 1. I = ln 12 7 2. I = ln 5 2 3. I = 1 2 ln 12 7 4. I = 2 ln 5 2 correct 5. I = 1 2 ln 5 2 6. I = 2 ln 12 7 Explanation: Integrating the inner integral with respect to x keeping y fixed, we see that Z 4 2 ( x + y ) 2 dx = h- 2 x + y i 4 = 2 n 1 y- 1 4 + y o . In this case I = 2 Z 4 1 n 1 y- 1 4 + y o dy = 2 h ln y- ln(4 + y ) i 4 1 . Consequently, I = 2 ln ‡ (4)(1 + 4) (4 + 4) · = 2 ln 5 2 . keywords: iterated integral, rational function, log integral 002 (part 1 of 1) 10 points Evaluate the iterated integral I = Z ln 4 ˆ Z ln 3 e 2 x- y dx ! dy . 1. I = 4 2. I = 3 correct 3. I = 5 4. I = 2 5. I = 6 Explanation: Integrating with respect to x with y fixed, we see that Z ln 3 e 2 x- y dx = 1 2 h e 2 x- y i ln 3 = 1 2 ‡ e 2 ln 3- y- e- y · = ‡ 3 2- 1 2 · e- y . Thus I = 4 Z ln 4 e- y dy =- 4 h e- y i ln 4 =- 4 ‡ e- ln 4- 1 · . Consequently, I =- 4 ‡ 1 4- 1 · = 3 . keywords: 003 (part 1 of 1) 10 points Kim, Jin – Homework 9 – Due: Oct 30 2007, 3:00 am – Inst: Diane Radin 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 18 · 2. I = 32 ln ‡ 13 9 · 3. I = 64 ln ‡ 9 13 · 4. I = 64 ln ‡ 13 18 · 5. I = 32 ln ‡ 9 13 · 6. I = 64 ln ‡ 13 9 · correct Explanation: The double integral over the rectangle A can be represented as the iterated integral I = Z 2 µZ 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 ) = 3 xe 3 xy over the rectangle A = { ( x, y ) : 0 ≤ x ≤ 3 , ≤ y ≤ 1 } . 1. I = 1 6 ‡ e 9- 8 · 2. I = 1 3 ‡ e 9- 10 · correct 3. I = 1 6 ‡ e 9- 9 · 4. I = 1 3 ‡ e 9- 9 · 5. I = 1 6 ‡ e 9- 10 · 6. I = 1 3 ‡ e 9- 8 · Explanation: The integral is given by I = Z Z A 3 xe 3 xy dxdy. Since the integral with respect to y can be evaluated easily using substitution (or di- rectly making the substitution in one’s 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 =...
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HW 9 Solution - Kim Jin – Homework 9 – Due 3:00 am –...

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