Question

To determine the common volume for two spheres.

Short answer

The common volume for two spheres is \(\frac{5}{{12}}\pi {r^3}\).

Step-by-step solution

Given

The sphere with radius \(r\). if the centre of each sphere lies on the surface of the other sphere.

The Concept ofcircle and the volume of the solid

The Equation of the circle is\({\left( {x + \frac{r}{2}} \right)^2} + {y^2} = {r^2}\)

The expression to find the volume of the solid is\(V = \int_a^b A (x)dx\)

Evaluate the volume of the solid

Consider the Equation of the circle \({\left( {x + \frac{r}{2}} \right)^2} + {y^2} = {r^2}\) ……. (1)

The solid is obtained by rotation of the area common to two circles of radius\(r\).

Sketch the two spheres intersects as shown in Figure 1.

Find the Area of the circle.

\(\begin{aligned}{}A(x) = \pi {y^2}\\ = \pi \left( {{r^2} - {{\left( {x + \frac{r}{2}} \right)}^2}} \right)\end{aligned}\)

The expression to find the volume as shown below.

\(V = \int_{ - r}^r A (x)dx\) ……… (2)

Substitute \( - r\) for a, r for \(b\), and \(\pi \left( {{r^2} - {{\left( {x + \frac{r}{2}} \right)}^2}} \right)\) for \(A(x)\) in Equation (2).

\(V = 2\pi \left( {{r^2} \times \frac{r}{2} - \frac{1}{3}{{\left( {\frac{r}{2} + \frac{r}{2}} \right)}^3} - \left( {0 - \frac{1}{3}{{\left( {0 + \frac{r}{2}} \right)}^3}} \right)} \right)\)

Further we get

\(\begin{aligned}{}V = 2\pi \left( {\frac{{{r^3}}}{2} - \frac{{{r^3}}}{3} + \frac{{{r^3}}}{{24}}} \right)\\ = 2\pi {r^3}\left( {\frac{{12 - 8 + 1}}{{24}}} \right)\\ = 2\pi {r^3}\left( {\frac{5}{{24}}} \right)\\ = \frac{{5\pi {r^3}}}{{12}}\end{aligned}\)

Therefore, the volume common to two spheres is \(\frac{5}{{12}}\pi {r^3}\).