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#?

Question

Isaiah Allen · Accepted Answer

#V = (4pi)/3#

Looking at the graph of #y = 2 - 2x#, 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 - 2x#.

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 - 2x#. 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*(f(x))^2 dx#

where #f(x)# is the curve we're revolving about the x-axis, and #[a,b]# is the interval we're concerned with. Important to note is the #pi*(f(x))^2# - this just means "area of a circle with radius #f(x)#." Multiplying this by #dx# gives the volume of a cylinder (disk) of width #dx# and radius #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(x) = 2 - 2x#, and #[a,b] = [0,1]#, and substitute:

#V = int_0^1 pi*(2 - 2x)^2 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 = (4pi)/3#

Paisley Johnson · Answer

undefined To find the volume of the solid generated by revolving the area bounded by the curves $x=0$, $y=0$, and $y=-2x+2$ about the x-axis using the disk method, follow these steps:

1. Determine the limits of integration by finding the intersection points of the curves. Set $y = -2x + 2$ equal to $y = 0$ to find the upper limit of integration.
   $$0 = -2x + 2$$
   $$x = 1$$

2. The lower limit of integration is $x = 0$, as given.

3. The radius of each disk is the distance from the curve to the axis of revolution. In this case, it is the y-coordinate of the curve. So, the radius $r$ is given by $r = y$.

4. The infinitesimal volume $dV$ of each disk is calculated using the formula for the volume of a cylinder, $dV = \pi r^2 dx$.

5. Integrate $dV$ from the lower limit of integration ($x = 0$) to the upper limit ($x = 1$) to find the total volume:
   $$V = \int_{0}^{1} \pi (-2x + 2)^2 \, dx$$

6. Simplify and solve the integral to find the volume of the solid.

Thus, by integrating the expression $ \pi (-2x + 2)^2 $ from 0 to 1, the volume of the solid generated by revolving the given area about the x-axis can be determined using the disk method.