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: $\int \pi\ x f(x) dx$ ??
Finding the volume when a region is rotated about the $y$-axis?
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calculus
graphing-functions
algebraic-curves
rotations