http://www.ck12.org Chapter 5. Applications of Definite Integrals
Figure 18
Solution:
Sincey=x^3 / 2 ,
dy
dx=3
2 x1 / (^2).
Using the formula above, we get
∫b
a
√
1 +[f′(x)]^2 dx=∫ 3
1√
1 +
[ 3
2 x1 / 2] 2
dx=∫ 3
1√
1 +^94 xdx.Usingu−substitution by lettingu= 1 +^94 x, thendu=^94 dx.Substituting, and remembering to change the limits of
integration,
L=^49
∫ 31 / 4
13 / 4√udu= 278
[
u^3 /^2] 31 / 4
13 / 4
≈ 4. 65.