The x value forms the radius of the solid. The maximum value of x is when x=√(π/b), x2=π/b=π/4, so y=asin(bx2)=8sin(π)=0. x=√π/2.
x is the radius of the base of the solid, so the circumference of the base, c=2πx=π√π.
The height is h=a=8, because sine has a maximum value of 1 and the curve rises to its maximum somewhere between the limits of x=0 and x=√π/2 (y=0 when x=0).
The shell method consists of summing the volumes of infinitesimally thin cylindrical shells of height y, radius x and thickness dx. Each has a volume 2πxydx=2πx(8sin(4x2)dx=16πxsin(4x2)dx.
The volume V of the solid is 2π0∫√π/28xsin(4x2)dx. If u=-cos(4x2), du=8xsin(4x2)dx, and the limits for u are -cos(0)=-1 when x=0 and -cos(π)=1 when x=√π/2. V=2π -1∫1du=2π[u]-11=2π(1-(-1))=4π.