Volume, disc and shell method

[ayrestaurant]

Find the volume generated by revolving the region bounded by y=18x-3x^2 and y=9x about the y-axis.
a. pi/12 b. pi/6 c. 3pi/4 d. pi/3

 

[galactus]

Find the volume generated by revolving the region bounded by \(\displaystyle y=18x-3x^{2}\) and \(\displaystyle y=9x\) about the y-axis.
\(\displaystyle a. \;\ \frac{\pi}{12}\)

\(\displaystyle b. \;\ \frac{\pi}{6}\)

\(\displaystyle c. \;\ \frac{3\pi}{4}\)

\(\displaystyle d. \;\ \frac{\pi}{3}\)

Since we are revolving about the y-axis, we can use shells.

With shells, the cross sections are parallel to the axis about which we are revolving.

Thus, we integrate w.r.t x

The functions intersect at \(\displaystyle 18x-3x^{2}=9x\)

\(\displaystyle x=0, \;\ x=3\)

\(\displaystyle 2\pi\int_{0}^{3}x\left(18x-3x^{2}-9x\right)dx\)

Even using washers, which is more complicated, gives the same result.

\(\displaystyle \pi\int_{0}^{27}\left[\left(\frac{y}{9}\right)^{2}-\left(\frac{-\sqrt{81-3y}+9}{3}\right)^{2}\right]dy\)

This is why it is best to use one method over the other. In this case, shells is preferable because of the

easier integration.

The solution does not match any of your choices.

Perhaps I am misinterpreting the problem. Did you type it correctly?. Make sure. Easy to make typos.