In addition to finding the volume of unusual shapes, integration can help you to derive volume formulas. For example, you can use the disk/washer method of integration to derive the formula for the volume of a cone.

Integration works by cutting something up into an infinite number of infinitesimal pieces and then adding the pieces up to compute the total. The disk/washer method cuts up a given shape into thin, flat disks or washers; this makes it useful for shapes with circular cross-sections, like, well, cones.

So the cone's volume is exactly one third ( 1 3) of a cylinder's volume. (Try to imagine 3 cones fitting inside a cylinder, if you can!) Volume of a Sphere vs Cylinder. Now let's fit a cylinder around a sphere. We must now make the cylinder's height 2r so the sphere fits perfectly inside. Volume of a cone. March 2, 2019 March 2, 2019 Craig Barton. Author: Rebecca Hall. This type of activity is known as.

The following practice question asks you to apply the disk method for just this purpose.

Practice question

1. Use the disk method to derive the formula for the volume of a cone. Hint: What’s your function? See the following figure. Your formula should be in terms of r and h.

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Answer and explanation

1. The formula is

   How do you get it? First, find the function that revolves about the x-axis to generate the cone.

   The function is the line that goes through (0, 0) and (h, r). Its slope is thus

   and its equation is therefore

   Now express the volume of a representative disk. The radius of your representative disk is f (x) and its thickness is dx. Thus, its volume is given by

   Finally, add up the disks from x = 0 to x = h by integrating. (Don’t forget that r and h are constants that behave like numbers.)

When you want to compute the volume of a cone, you need only two things: its height and the radius of its base. Even if you are given its slant height instead of its vertical height, you can still find the volume; you just need to include an extra step.

A cone is a solid figure with a rounded base and a rounded lateral surface that connects the base to a single point.

A right circular cone is a cone with a circular base, whose peak lies directly above the center of the base. Most cones in geometry books are right circular cones.

Here’s how to find the volume of a cone.

Now for a cone problem:

Here’s the diagram proof.

To compute the cone’s volume, you need its height and the radius of its base. The radius is, of course, half the diameter, so it’s

Then, because the height is perpendicular to the base, the triangle formed by the radius, the height, and the slant height is a 30-60-90 triangle. You can see that h is the long leg and r the short leg, so to get h, you multiply r by the square root of 3:

You’re now ready to use the cone volume formula:

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