Chapter 5 - Applications Of The Definite Integral In Geometry, Science, And Engineering - 5.3 Volumes By Cylindrical Shells - Exercises Set 5.3 - Page 370: 8

$$V = \frac{{\sqrt 2 }}{2}\pi $$

$$\eqalign{ & {\text{We have }}y = \cos \left( {{x^2}} \right),\,\,\,x = 0,\,\,\,x = \frac{1}{2}\sqrt \pi ,\,\,y = 0 \cr & {\text{The volume of the solid can be calculated using cylindrical shells}} \cr & V = \int_a^b {2\pi x\left[ {f\left( x \right) - g\left( x \right)} \right]dx} \cr & {\text{Let }}f\left( x \right) = \cos \left( {{x^2}} \right){\text{ and }}g\left( x \right) = 0 \cr & V = \int_0^{\frac{1}{2}\sqrt \pi } {2\pi x\left( {\cos \left( {{x^2}} \right) - 0} \right)dx} \cr & V = \pi \int_0^{\frac{1}{2}\sqrt \pi } {\cos \left( {{x^2}} \right)\left( {2x} \right)dx} \cr & {\text{Integrating}} \cr & V = \pi \left[ {\sin \left( {{x^2}} \right)} \right]_0^{\frac{1}{2}\sqrt \pi } \cr & V = \pi \left[ {\sin {{\left( {\frac{1}{2}\sqrt \pi } \right)}^2} - \sin {{\left( 0 \right)}^2}} \right] \cr & V = \pi \left[ {\frac{{\sqrt 2 }}{2} - 0} \right] \cr & V = \frac{{\sqrt 2 }}{2}\pi \cr} $$