Is mulitvariable calculus unavoidable when finding the formula of a volume the cylinder by integration?

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I am not in multivariable calculus yet, but I am going crazy trying to create the formula for a cylinder by integration, not a homework assignment. I can't put h in terms of r; is there another way to do this using single variable calculus? If the answer is no, then is the inability to put one variable in terms of the other mean that I must use multivariable calculus?

Edited by Science Student

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The radius and height are quite independent, unless you want to put some constraints here like fixing the surface area. Anyway, you don't really need multivariable calculus if you realise that the cylinder is an interval times a circle.

However, if you want to start from a volume form and integrate it then you do need multivariable calculus, but it is not difficult as my above comments suggest.

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The radius and height are quite independent, unless you want to put some constraints here like fixing the surface area. Anyway, you don't really need multivariable calculus if you realise that the cylinder is an interval times a circle.

However, if you want to start from a volume form and integrate it then you do need multivariable calculus, but it is not difficult as my above comments suggest.

Is the interval a fixed height?

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A cylinder is what is called a volume of revolution or solid of revolution.

You get a VOR when you swing a curve (straight line in this case) about a coordinate axis.

So a cone, a cylinder, a greek urn, a lampstand or anything you might turn on a lathe are solids of revolution.

The volume can be calculated by ordinary single variable calculus.

If we rotate the curve f(x) about the x axis the volume from x= a to x = b is given by

$V = \int\limits_a^b {\pi {{(f(x))}^2}} dx$

Edited by studiot

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A cylinder is what is called a volume of revolution or solid of revolution.

You get a VOR when you swing a curve (straight line in this case) about a coordinate axis.

So a cone, a cylinder, a greek urn, a lampstand or anything you might turn on a lathe are solids of revolution.

The volume can be calculated by ordinary single variable calculus.

If we rotate the curve f(x) about the x axis the volume from x= a to x = b is given by

$V = \int\limits_a^b {\pi {{(f(x))}^2}} dx$

Thanks!

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