Volume is understood as length times width times height, or on a graph, x times y times z. When an integral is revolved around an axis, this is the area.

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Presentation transcript:

Volume is understood as length times width times height, or on a graph, x times y times z. When an integral is revolved around an axis, this is the area of the cross section times the change in x or y (dx or dy).

If revolved around the y-axis, cross-sections perpendicular to the y-axis are in the shape of disks. The area of the cross-section is the area of a circle, where the radius is the parabolic equation if solved for x. This is then integrated over the interval.

For washer shaped cross-sections the integral for the volume of the inner circle is subtracted from the integral for the volume of the outer circle.

Integrals work the same way as regular volumes. Finding the volume of y=x from 0 to 5, revolved around the x-axis will find the same volume as the formula for the volume of a cone.

This is a very useful topic in calculus because you can find the volume of any object for which you know the equation for its radius. (like if you know the volume of a jelly bean jar and the volume of a jelly bean)

Hello, I am Tara Conlon. Some people think I’m crazy. I don’t think so. The people in the ‘hospital’ said that I’m just different. So there, ha ha ha!