Question

Use the method of cylindrical shells to determine the volume of the solid obtained by rotating the region bound by y=2x+1, y=3, and x=4 about the line y = 10. (please make sure to use SHELL method and NOT disk method) thank you :)

Answer 1

We can plot the region and axis of rotation as follows

40- (4, 9) -5 (4, 3) 5 0 LO LO

The shell radius is given by

R= 10 y

The shell height is given by

$H = 4 - \frac{y-1}{2}$

The volume of rotation using the shell method about the horizontal line is given by

R.H.dy V = 2T

$V = 2\pi \int_{3}^{9} (10-y)\left ( 4 - \frac{y-1}{2} \right )dy$

$V = 2\pi \int_{3}^{9} (10-y)\left ( \frac{8-y+1}{2} \right )dy$

$V = 2\pi \int_{3}^{9} (10-y)\left ( \frac{9-y}{2} \right )dy$

$V = \pi \int_{3}^{9} (10-y)(9-y)dy$

After integrating we get,

$V =90\pi$

I hope this answer helps,
Thanks,
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