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Finding the Volume of a Solid with Similar Cross Sections

How to Evaluate the Volume of a Solid of Revolution

You can evaluate the volume of a solid of revolution. A solid of revolution is created by taking a function, or part of a function, and spinning it around an axis — in most cases, either the x-axis or the y-axis.

A solid of revolution of <i>y</i> = 2 sin <i>x</i> around the <i>x</i>-axis.
A solid of revolution of y = 2 sin x around the x-axis.

For example, the left side of the figure shows the function y = 2 sin x between x = 0 and

image1.png

Every solid of revolution has circular cross sections perpendicular to the axis of revolution. When the axis of revolution is the x-axis (or any other line that’s parallel with the x-axis), you can use the meat-slicer method directly.

However, when the axis of revolution is the y-axis (or any other line that’s parallel with the y-axis), you need to modify the problem.

To find the volume of this solid of revolution, use the meat-slicer method:

  1. Find an expression that represents the area of a random cross section of the solid (in terms of x).

    This cross section is a circle with a radius of 2 sin x:

    image2.png
  2. Use this expression to build a definite integral (in terms of dx) that represents the volume of the solid.

    This time, the limits of integration are from 0 to π/2:

    image3.png
  3. Evaluate this integral by using the half-angle formula for sines:

    image4.png

    Now evaluate:

    image5.png

So the volume of this solid of revolution is approximately 9.8696 cubic units.

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