The volumes of many solids can be obtained by the application of the slice method. In the previous section, we sliced the solid into infinitely many thin disks. However, the element of volume does not have to be necessarily a disk (or a washer).

In a more general case, imagine the solid is sliced into infinitesimally thin slices of thickness \(dx\) by a family of planes perpendicular to the \(x\)-axis (Figure 1). Suppose you know a formula for the area of an arbitrary cross-section of the solid made by such planes \(A(x)\). Some common cross-sections are triangles, squares, rectangles, trapezoids, and semicircles. Then the volume of a slice (the element of volume) is this area \(A(x)\) multiplied by the thickness of the slice \(dx\): \[dV=A(x)dx.\] The total volume of the solid is the sum of the volumes of these slices. If the solid is bounded by two parallel planes perpendicular to the \(x\)-axis at \(x=a\) and \(x=b\), then \[\boxed{V=\int_{a}^{b}dV=\int_{a}^{b}A(x)dx.}\]

*Figure 1 *

Similarly, suppose that the solid is cut by planes perpendicular to the \(y\)-axis and \(A(y)\) is the area of an arbitrary cross section (Figure 2). Then \[\boxed{V=\int_{c}^{d}A(y)dy}\] if the solid is bounded between the planes \(y=c\) and \(y=d\) (\(c<d\)).

*Figure 2 *