Question

Consider the following graph:
http://i55.tinypic.com/288nvpz.gif

a=4 and b=5
Refer to the figure and find the volume V generated by rotating the region R1 about the line AB.

Answer 1

R1 is region bounded by y = 4x^5, y=0, and x=1
Rotating around vertical axis x=1 means we will look at horizontal cross-sections which are circular with radius that is x-distance from 4x^5 and x=1
Also we will integrate w/ respect to y since each horizontal cross-section has height of dy
get x in terms of y
y=4x^5 --> x = (y/4)^1/5
radius = 1 - (y/4)^1/5
V = pi*integral 0-4 R^2 dy
R^2 = (1-(y/4)^1/5)^2 = 1-2(y/4)^1/5 +(y/4)^2/5
\[V = \pi \int\limits_{0}^{4}(y/4)^{2/5} -2(y/4)^{1/5}+1 dy\]
u=y/4 ---> du =dy/4 -->4du = dy
\[V = 4\pi \int\limits_{0}^{4}u ^{2/5} - 2u ^{1/5}+1 du \]
\[V=4\pi |_{0}^{4} \left( 5/7(y/4) ^{7/5}- 5/3(y/4) ^{6/5} +y/4 \right)\]
\[V =4\pi/21\]