Find the volume of solid of revolution obtained when the region bounded by the graph of y = sin(x^2), the x-axis, and the lines x = 0, x = sqrt(pi) , is revolved about the y-axis. Page

Answer 1

We will use the "shell method" to solve this problem. The thickness of each cylindrical shell is dx, the circumference is 2πx, and the height is sin(x²). Therefore, the volume of each shell dV is:

dV=2πxsin(x²)dx

To find the volume V of the entire solid of revolution we will add up the volumes dV of all the shells located between x=0 and x=√π:

V=integral from 0 to √π of 2πxsin(x²)dx=-πcos(x²) from 0 to √π=-πcos[(√π)²]+πcos(0²)=-πcos(π)+πcos(0)

=-π*-1+π*1=π+π=2π

Therefore, the volume of this solid of revolution is 2π.