Math 166Z Homework # 2
Due Tuesday, January 27th
Please do not try to fit your answers onto this page. They will not fit. No credit will be
given unless you show your work.
1. Sketch the region bounded by the given curves, then use the disk method to find the
volume of the solid obtained by rotating the region about the given line.
(a) y = x2 , x = 1 , y = 0 ; about the x-axis
(b) x = y − y 2 , x = 0 ; about the y-axis
√
(c) y = 3 x , x = 4y
i. about the x-axis
ii. about the line x = 8
iii. about the line y = 3
2. Sketch the region bounded by the given curves, then use the method of cylindrical
shells to find the volume of the solid obtained by rotating the region about the given
line.
√
(a) y = sin(x2 ) , y = 0 , x = 0 , x = π ; about the y-axis
(b) y 2 = x , x = 2y ; about the y-axis
√
(c) x = 4 y , x = 0 , y = 16 ; about the x-axis
√
(d) y = x − 1 , y = 0 , x = 5 ; about y = 3
(e) y = 4x − x2 , y = 8x − 2x2 ; about x = −2
3. Sketch the region bounded by the given curves, then use any method you would like
to find the volume of the solid obtained by rotating the region about the given line.
(a) y = x2 + x − 2 , y = 0 ; about the x-axis
(b) x2 + (y − 1)2 = 1 ; about the y-axis
(c) y = x2 − x3 , y = 0 ; about the y-axis
4. Each of the following integrals represents the volume of a solid. Describe the solid or
provide a sketch.
Z 1
(a) π
(y − y 2 ) dy
Z
0
π/2
(b)
2πx cos(x) dx
0
5. Find the length of the given curve:
3
(a) y = 13 (x2 + 2) 2 ; 0 ≤ x ≤ 1
(b) y =
x4
4
+
1
8x2
; 1≤x≤3
(c) y =
x3
6
+
1
2x
; 1≤x≤2
Bonus Question: (worth 5 point extra credit)
2
Consider the area of the region between the function y = ex (i.e. y = exp(x2 ) in case you can’t read it)
and the x-axis from x=0 to x=2.
(a) Find the volume of the solid obtained by revolving this region about the y-axis.
(b) Set up the integral for finding the volume of the solid obtained by revolving this region
about the line y = 2. Do not evaluate this integral (what problems would you have?)
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