8.9 Volume with Disc Method: Revolving Around the x- or y-Axis

The disc method is used when a region is revolved around an axis and the resulting cross sections perpendicular to that axis are solid circles with no hole. If the radius of a typical disc is r, then its area is ${\pi r^2}$, so the volume is found by integrating ${\pi r^2}$. When revolving around the x-axis, you usually integrate with respect to x if the function is given as y=f(x). When revolving around the y-axis, you usually integrate with respect to y if the function is given as x=g(y).

The most important step is identifying the radius correctly. The radius is always the distance from the axis of rotation to the curve. If the region touches the axis, then there is no inner hole, so discs are the right method. This lesson is the rotational analogue of cross sections: instead of a given shape like a square or triangle, the rotation itself creates circular cross sections. The volume formulas come from adding up the areas of those circles across the interval.