Find circumference, volume, and height of y=a sin(bx^2) using shells method

Question

Let S be the solid obtained by rotating the region shown in the figure about the y-axis. (Assume a = 8 and b = 4.)

y=a sin(bx^2) on interval [0, sqrt pi/b]

 

-Find its circumference c and height h.

 

-Use shells to find the volume V of S.

Answer 1

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.