We define a region \(R\), bounded above by the graph of a function \(y=f(x)\), below by the x-axis, and on the left and right by the lines \(x=a\) and \(x=b\). We then revolve this region around the y-axis:
We can partition the interval \([a,b]\) using a regular partition, \(P={x_0,x_1,…,x_n}\) and, for \(i=1,2,…,n\), let \(x^*_i\) be the midpoint of the interval \([x_{i-1},x_i]\). Then, construct a rectangle over the interval \([x_{i-1},x_i]\) of height \(f(x^*_i)\) and width \(\Delta x\) (figure \((a)\) below). When that rectangle is revolved around the y-axis, we get a cylindrical shell:
The volume of one shell in the interval \([x_{i-1},x_i]\) is the volume of the outer cylinder \((\pi x^2_i f(x_i))\) minus the volume of the inner cylinder \((\pi x^2_{i-1} f(x_{i-1}))\):
Since \(x^2_i - x^2_{i-1} = (x_i + x_{i-1})(x_i - x_{i-1})\):
Another way to think of this is to think of making a vertical cut in the shell and then opening it up to form a flat plate:
To calculate the volume of the entire solid, we then add the volumes of all the shells:
Taking the limit as \(n→∞\):