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# Quiz 1 Key - Math 262 Quiz 1 Solutions Name This is a...

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Math 262 Quiz 1 Name: Solutions This is a Take-Home Quiz. You may use your book and course notes, and you may consult with other members of the class, but you may not consult with outside tutors (at least not on these speciFc problems). 1. (5 points) ±ind the arc length of the curve given by y = x 2 3 on [0 , 4]. Recall that the basic formula for Fnding arc length is: L = i b a r 1 + [ f ( x )] 2 dx Here, f ( x ) = 2 3 x 1 3 , so we get: L = i 4 0 R 1 + b 2 3 x 1 3 B 2 dx = i 4 0 R 1 + 4 9 x 2 3 dx = i 4 0 R 1 + 4 9 x 2 3 dx We continue to simplify in order to get to a form where we are able to evaluate the integral. = i 4 0 ± ² ² ³ 9 x 2 3 + 4 9 x 2 3 dx = i 4 0 r 9 x 2 3 + 4 r 9 x 2 3 dx = i 4 0 r 9 x 2 3 + 4 3 x 1 3 dx = i 4 0 1 3 x 1 3 r 9 x 2 3 + 4 dx We now substitute using u = 9 x 2 3 + 4 and du = 6 x 1 3 dx This gives 1 6 i 9(4 2 3 )+4 4 1 3 u 1 2 du = 1 18 3 2 u 3 2 v v v v v 9(4 2 3 )+4 4 Evaluating this gives: 1 27 b p 9 p 4 2 3 P + 4 P 3 2 - 8 B 2. (5 points) ±ind the area of the surface generated by rotating the curve y = x 3 between x = 0 and x = 2 about the x -axis. Recall that the basic formula for Fnding surface area in this situation is:
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