Question

To determine the volume of the remaining portion of the sphere.

Short answer

The volume of the portion which rremains of the sphere is \(\frac{{4\pi }}{3}{\left( {{R^2} - {r^2}} \right)^{\frac{3}{2}}}\).

Step-by-step solution

Given

The radius of sphere is denoted as \(R\).

The radius of hole is denoted as \(r\).

The Concept of area and the volume of the solid

The area of the region which rotatesis

\(\begin{aligned}{c}A(y) = \pi \left( {{{\left( {\sqrt {{R^2} - {x^2}} } \right)}^2} - {r^2}} \right)\\ = \pi \left( {{R^2} - {x^2} - {r^2}} \right)\end{aligned}\)

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

Evaluate the volume of the solid

Sketch the sphere with hole as shown in Figure

Refer to Figure 1

The intersection point of line \(y = r\) makes a semi-circle.

\(\begin{aligned}{c}B{C^2} = A{B^2} - A{C^2}\\{r^2} = {R^2} - {x^2}\\{x^2} = {R^2} - {r^2}\\x = \pm \sqrt {{R^2} - {r^2}} \end{aligned}\)

Sketch the intersection points as shown in Figure

Refer to Figure 2

The rotating shaded region about the \(x\)-axis gives the volume.

Find the area of the rotating region as shown below.

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

Find the volume of the rotating region as shown below.

\(V = \int A (y)dx\) ………. (1)

Substitute \(\pi \left( {{R^2} - {x^2} - {r^2}} \right)\) for \(A(y)\) in Equation (1).

\(\begin{aligned}{}V = \int_{ - \sqrt {{R^2} - {r^2}} }^{\sqrt {{R^2} - {r^2}} } \pi \left( {{R^2} - {x^2} - {r^2}} \right)dx\\ = 2\pi \int_0^{\sqrt {{R^2} - {r^2}} } {\left( {{R^2} - {x^2} - {r^2}} \right)} dx\\ = 2\pi \left( {{R^2}x - \frac{{{x^3}}}{3} - {r^2}x} \right)_0^{\sqrt {{R^2} - {r^2}} }\\ = 2\pi \left( {\left( {{R^2} - {r^2}} \right)\sqrt {{R^2} - {r^2}} - \frac{{{{\left( {\sqrt {{R^2} - {r^2}} } \right)}^3}}}{3} - 0} \right)\end{aligned}\)

Solve further we get

\(\begin{aligned}{}V = 2\pi \left( {{R^2} - {r^2}} \right)\sqrt {{R^2} - {r^2}} \left( {1 - \frac{1}{3}} \right)\\ = \frac{{4\pi }}{3}{\left( {{R^2} - {r^2}} \right)^{\frac{3}{2}}}\end{aligned}\)

Therefore, the volume of the portionwhich rremains of the sphere is \(\frac{{4\pi }}{3}{\left( {{R^2} - {r^2}} \right)^{\frac{3}{2}}}\).