# Finding the volume when a region is rotated about the $y$-axis?

My basic question is: When we think of the area under the graph and extending it in the 3 dimensions, we actually get a cylinder with height = $f(x)$, thickness = $dx$ and inner radius = $x$. Then the volume should be $\int \pi (2x+dx) \ dx\ f(x)$. Right? Then why is it: $\int \pi\ x f(x) dx$ ??

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You just need to think carefully about what is going on with the method of shells. I suggest reading here (scroll down about halfway where they start talking about shells).

Another great resource that explains what you need is video #29 in this series. (The entire set of videos is excellent, with great visuals as well as step-by-step explanations.)

Most generally (and as the links above will explain), the method of shells produces an integral of the form $$\int_a^b \pi\cdot(\text{shell radius})\cdot(\text{shell height})\,dx,$$ i.e., $\int_a^b \pi\,x f(x)\,dx$ as you mentioned.

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