Find the volume common to two spheres, each with radius r, if the center of each sphere lies on the surface of the other sphere.
The volume common to two spheres
Let be a solid that lies between . If the cross sectional area of in the plane , through and perpendicular to the x-axis, is where is continuous function, then the volume of is
Draw a circle whose centre is at origin, that is (0, 0) and radius is . Draw another circle whose centre is at (r, 0) and radius , which is as shown below. Then volume common to both sphere is the shaded region whose volume can be calculated by calculating the volume of each shaded region and then add them or by just calculating the volume of one shaded region and then multiplying by , as volume of both shaded region is same.
Equation of circle with points is
For circle with origin, equation of circle is
For circle with point (r, 0), equation of circle is
Considering the shaded region of circle whose equation is
Solving for y,
So, by symmetry, the total volume is twice the volume obtained