Volume of a solid of revolution: method of disks

In the previous post we define the formula to find the volume of a solid which consists in integrating the area of a cross-section. In the following post I am going to find another version of this formula using the method of disks.

 Method of disks

Let f be a function continuous and positive in an interval [a,b]. Let’s consider a region R between f and the x-axis. Let’s R revolve about the x-axis. This action generates a solid of revolution with cross-circular sections with radii f(x) for any value of x.

Volume formula by the method of disks

In the previous lesson we learn that the volume of a solid is found by integrating the area of the cross-circular section within a definite interval.

Example

Calculate the volume of the solid that is obtained when the region under the curve x  is revolved about the x−axis over the interval [1,7].

Solution

Here is the graph of the function f:

We revolve the region between f(x) and the x-axis. We obtain the following figure:

Since the cross-section is circular, the volume of the solid is given by:

Practice
.
Demonstrate that the volume of a sphere with radius r is V = 4𝛑r³/3

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