The volume of a solid obtained by rotating the region bounded by the given curves about y-axis.
i. If x is the radius of a typical shell, then circumference=2πx and height is y
ii. By shell method, the volume of the solid by rotating the region under the curve y=f(x) about y- axis from a to b is
The region bounded by x2-y2=a2, x=a+h rotated about the y- axis (a>0, h>0).
The graph of x2-y2=a2 is a hyperbola with right and left branches.
Solving x2-y2=a2 for y,
Taking square root on both the sides,
y= ±x2- a2
Using shell method, find the typical approximating shell with radius x.
Therefore, the circumference is 2πx and height of each shell is
x2- a2 - - x2- a2=2x2- a2
From the figure, the limits of integration is from a to a+h.
So the total volume is
V= ∫aa+h2πx[2x2- a2] dx
Use substitution u=x2-a2.
Hence,du=2xdx , i.e. xdx=du2
For x=a, u=a2-a2=0 and
for x=a+h, u=a+h2-a2=a2+2ah+h2-a2=2ah +h2
Using this in the above integral, it becomes
The volume of the solid obtained by rotating the region bounded by the given curves