Volume of a hyperbolic cap Consider the region R bounded by the right branch of the hyperbola x2/a2 − y2/b2 = 1 and the vertical line through the right focus.

a. What is the volume of the solid that is generated when R is revolved about the x-axis?

b. What is the volume of the solid that is generated when R is revolved about the y-axis?

Solution 81EStep 1 :

Given : .a. To calculate the volume of the hyperbolic cap, we know that the volume of the solid generated by revolving the curve y=f(x) about the x-axis is given by

Here the volume generated by is calculated by putting and using the limit from vertex (a,0) to focus (ae,0) as follows

Hence the volume of the solid that is generated when R is revolved about the x-axis is : .