http://www.ck12.org Chapter 5. Applications of Definite Integrals
Figure 20
Area of a Surface of Revolution
Iffis a smooth and non-negative function in the interval[a,b],then the surface areaSgenerated by revolving the
curvey=f(x)betweenx=aandx=babout thex−axis is defined by
S=
∫b
a^2 πf(x)√
1 +[f′(x)]^2 dx=∫b
a^2 πy√
1 +
(dy
dx) 2
dx.Equivalently, if the surface is generated by revolving the curve about they−axis betweeny=candy=d,then
S=
∫d
c^2 πg(y)√
1 +[g′(y)]^2 dy=∫d
c^2 πx√
1 +
(dx
dy) 2
dy.Example 1:
Find the surface area that is generated by revolvingy=x^3 on[ 0 , 2 ]about thex−axis (Figure 21).
Solution: