User:Silver2Falcon
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One usage of the definite integral is to find the area of a cone, using the fundamental theorem of calculus.
First, we need the formula of the three dimensional shape--the forumula on a two dimensional plane that will create the 3d shape when rotated. For this example, we will use the formula y=3x, and rotate it around the 'x' axis to create a cone.
For a cylinder,
- V = πr2h
.
Since r=y and height=dx
- V = πy2dx
Therefore...
Therefore...
![V=\pi \int\limits_{0}^{3} [f(x)]^2\, dx](../../../math/8/c/f/8cf2e67df4bd6f79a21b3739a242ced3.png)
Soooooo....
![V=\pi \int\limits_{0}^{3} [3x]^2\, dx](../../../math/4/b/a/4ba9243f332346d6144c02ddec988a07.png)
![V=\pi [9x^2] \bigg| ^3 _0](../../../math/4/3/5/435edd130bb7a1290928f79059e56112.png)
- V = π(3) − π(0)
Which means that the cone made when y=3x is rotated around the x axis has a volume of...
- V = 3πunits3
