[Math] Finding volume of a cone through integration

applicationsintegration

I am trying to find the volume of a cone using integration through horizontal slicing. The cone has a base radius of 10cm and a height of 5cm.

I am assuming this means I should integrate with respect to y, but I am not entirely sure how to set this up. I know that volume of a cylinder is given by the following:
$$V = \pi r^2h$$

So I am assuming that the integral would be:
$$\pi \int_0^5 f(y)^2dy$$

I am not sure how the x value of the radius 10cm (since it is not with respect to y) should fit into the equation, though.

Also, sorry for the pseudo-code style. I do not know how to use the math typesetting yet.

Best Answer

Place your cylinder so the center of the base is at $(0,0)$, and the apex is at $(0,5)$.

If you imagine looking at the cylinder straight on, it will look like a triangle with base $20$ and height $5$. If you make a horizontal slice at level $y$, then you get a figure with two similar triangles:

                   ^                  ^
                  / \                 | 
                 /   \                |
    ^           /_____\               5 
    |          /   2r  \              |
    y         /         \             |
    |        /           \            |
    V       /_____________\           V

            |----- 20 -----|

then using similar triangles note that the height of the triangle on the top is $5-y$, and the base is $2r$. So we have $$\frac{2r}{20} = \frac{5-y}{5}.$$ From this, we can express $r$ in terms of $y$.

This is where you are using the fact that the base of your original cone is $10$ cm.

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