The disc method is a way to find the volume by rotating around the x- or y-axis. In this situation, we will find the volume by adding up a bunch of infinitely thin circles.
Let’s look at the region between the curve y = √x and the x-axis from x = 0 and x = 1.
If you rotate this region around the x-axis, the cross sections will be circles with radii √x. Thus, the area of each cross section will be π(√x)^2 or πx. Now we can integrate πx from x = 0 and x = 1 to get the volume.
Now, let’s generalize this. If you have a region whose area is bounded by the curve y = f(x) and the x-axis on the interval [a,b], each disk has a radius of f(x), and the area of the disk will be
To find the volume, evaluate the integral.
Now, you try to use the formula with this example problem:
If rotate the function y = x + 2 about the x-axis from x = 0 to x = 2, what is the volume of the figure?
Solution: In this example, your function is y = x + 2 which looks like this:
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8.9Volume with Disc Method: Revolving Around the x- or y-Axis