Calc1.84

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Find the volume of a cylinder with radius 3 and height 10.

Here, we are finding the volume of an object with circles for cross-sections. It is quite simple since, in a cylinder, the circles are all the same size. So, the cross-sectional area at any point is just A=\pi (3)^{2}. Since the area does not depend on x, it does not matter which interval we integrate over so long as it has a length of 10. So the volume is

\int _{{0}}^{{10}}9\pi \,dx=9\pi x{\big |}_{{0}}^{{10}}=90\pi \,

Thus, the volume is 90\pi cubic units. This is the same answer we would get if we used the formula for the volume of a cylinder, V=\pi r^{2}h. This example shows that you can get the volumes of already known shapes but you can also get volumes for shapes with varying cross-sectional areas for which there is on specific formula.


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