Unformatted text preview: Sketch the region and a typical shell.
3
not have to evaluate the2expression.)
29. y
x 5 dx
0, y 1, y 2 n = 10
you do 0 , y 1 x
x 30. 2 0
0 y
4
■ 31. y 32. y 1
■ 2
■ ■ ■ ■ ■ 1 ■ ■ 0 ■ dy 1 y2 3 2
4 0 y 2 0 y 6 ■ y y1 2 y2 dy x cos x
■ ■ ■ sin x d x
■ ■ ■ ■ ■ ■ ■ ■ (b) If the region is instead rotated about the y axis, use the Midpoint Rule with n = 5 to estimate
the volume of the resulting solid. (Again, you do not have to evaluate your expression.) Math 42, Winter 2010 First Exam — January 28, 2010 Page 9 of 11 6. (10 points)
(a) Set up two distinct integrals, each in terms of a single variable, representing the area of the
√
region in the ﬁrst quadrant bounded by the curves y = x3 and y = x. For each, justify your
answer by drawing a picture and marking a sample slice. Don’t evaluate either integral. (b) Set up two distinct integrals, each in terms of a single variable, representing the volume of the
solid obtained by rotating the region from part (a) around the line x = 1. For each, make sure
you justify your answer (draw a picture, label a sample slice, and cite the method used). Don’t
evaluate either integral. Math 42, Winter 2010 First Exam — January 28, 2010 Page 10 of 11 7. (8 points) A napkin ring is made by taking a...
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This note was uploaded on 07/31/2013 for the course MATH 42 taught by Professor Butscher,a during the Spring '07 term at Stanford.
 Spring '07
 Butscher,A
 Math, Calculus

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