# Using the disk method, how do you find the volume of the solid generated by revolving about the x-axis the area bounded by the curves x=0, y=0 and y=-2x+2?

Sep 11, 2015

$V = \frac{4 \pi}{3}$

#### Explanation: Looking at the graph of $y = 2 - 2 x$, imagine the area between the line, the x-axis, and the y-axis, being revolved around the x-axis. You'll end up with a cone, with the point/tip at $x = 1$, and the center of the circular base (which has radius 2) at the origin.

Next, imagine looking at a cross-section parallel to the x-z axis - parallel to the circular base. Every cross-section will be a circle. At $x = 0$, the circle has radius 2, and at $x = 1$, the circle has radius 0. In fact, for any $x$, the cross-sectional circle has radius $2 - 2 x$.

To find the volume of the solid of revolution, we can imagine that our solid is composed of infinitely many disks, of infinitesimal width, and radius equal to $2 - 2 x$. To find the volume of the solid, we sum up (integrate) each disk. This process is commonly called the method of disks.

The general formula for the method of disks is:

$V = {\int}_{a}^{b} \pi \cdot {\left(f \left(x\right)\right)}^{2} \mathrm{dx}$

where $f \left(x\right)$ is the curve we're revolving about the x-axis, and $\left[a , b\right]$ is the interval we're concerned with. Important to note is the $\pi \cdot {\left(f \left(x\right)\right)}^{2}$ - this just means "area of a circle with radius $f \left(x\right)$." Multiplying this by $\mathrm{dx}$ gives the volume of a cylinder (disk) of width $\mathrm{dx}$ and radius $\mathrm{dx}$. And integration allows us to sum all these disks up, exactly what we want to accomplish.

So, in our case, we'll note that $f \left(x\right) = 2 - 2 x$, and $\left[a , b\right] = \left[0 , 1\right]$, and substitute:

$V = {\int}_{0}^{1} \pi \cdot {\left(2 - 2 x\right)}^{2} \mathrm{dx}$

Well, that was the difficult part; setting up the integral. From here, evaluating the integral should be fairly easy. I'll leave it to you as an exercise, but the answer should come out to

$V = \frac{4 \pi}{3}$