#### To determine

**To find:**

The volume common to two spheres

#### Answer

V=5r3π12

#### Explanation

**1) Concept:**

Let S be a solid that lies between x=a to x=b. If the cross sectional area of S in the plane Px, through x and perpendicular to the x-axis, is A(x), where A is continuous function, then the volume of S is

V=limn→∞∑i=1nA(xi*)∆x=∫abA(x)dx

**2) Calculation:**

Draw a circle whose centre is at origin, that is (0, 0) and radius is r. Draw another circle whose centre is at (r, 0) and radius r, 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 2, as volume of both shaded region is same.

Equation of circle with points (x1, y1) is

x-x12+y-y12=r2

For circle with origin, equation of circle is

x2+y2=r2

For circle with point (r, 0), equation of circle is

x-r2+y2=r2

Considering the shaded region of circle whose equation is x-r2+y2=r2

Solving for y,

y2=r2-x-r2

y=r2-x-r2

V=π∫0r2r2-x-r22dx

V=π∫0r2r2-x-r2dx

V=π∫0r2r2-x2-2rx+r2dx

V=π∫0r22rx-x2dx

V=π2rx22-x330r2

V=π2rr222-r233

V=π2rr28-r324

V=πr34-r324

V=π6r324-r324

V=5r3π24

So, by symmetry, the total volume is twice the volume obtained

Therefore,

Total volume 2V=25r3π24

Total volume=5r3π12

Volume common to two spheres is 5r3π12

**Conclusion:**

V=5r3π12