http://www.ck12.org Chapter 5. Applications of Definite Integrals
The volume of solid does not necessarily have to be circular. It can take any arbitrary shape. One useful way to find
the volume is by a technique called “slicing.” To explain the idea, suppose a solidSis positioned on thex−axis and
extends from pointsx=atox=b(Figure 6).
Figure 6
LetA(x)be the cross-sectional area of the solid at some arbitrary pointx.Just like we did in calculating the definite
integral in the previous chapter, divide the interval[a,b]intonsub-intervals and with widths
4 x 1 , 4 x 2 , 4 x 3 ,..., 4 xn.Eventually, we get planes that cut the solid intonslices
S 1 ,S 2 ,S 3 ,...,Sn.Take one slice,Sk.We can approximate sliceSkto be a rectangular solid with thickness 4 xkand cross-sectional area
A(xk).Thus the volumeVkof the slice is approximately
Vk≈A(xk) 4 xk.Therefore the volumeVof the entire solid is approximately
V=V 1 +V 2 +...+Vn
≈n
k∑= 1 A(xk)^4 xk.