Question

To determine the common volume for two circular cylinders.

Short answer

The common volume for two circular cylinders is \(\frac{{16}}{3}{r^3}\).

Step-by-step solution

Given

The circular cylinder with radius \(r\) if the axes of the cylinders intersect at right angles.

The Concept ofthe volume of the solid

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

Evaluate the volume

Sketch the two cylinders intersects at right angles with the cross-section as shown in Figure 1

Consider that the cross-section of the solid perpendicular to the \(x\)-axis is a square.

Show the calculation for area of the quarter square.

\(|PQ{|^2} = {r^2} - {x^2}\)

Calculate the area of the square.

\(A(x) = 4\left( {{r^2} - {x^2}} \right)\)

The expression to find the volume of the solid is shown below.

\(V = \int_a^b A (x)dx\) …………….. (1)

Substitute \( - r\) for a, r for \(b\) and \(4\left( {{r^2} - {x^2}} \right)\) for \(A(x)\) in Equation (1).

\(\begin{aligned}{}V = \int_{ - r}^r 4 \left( {{r^2} - {x^2}} \right)dx\\ = 8\int_0^r {\left( {{r^2} - {x^2}} \right)} dx\\ = 8\left( {{r^2}x - \frac{{{x^3}}}{3}} \right)_0^r\\ = 8\left( {{r^3} - \frac{{{r^3}}}{3} - 0} \right)\end{aligned}\)

Simplify further,

\(\begin{aligned}{}V = 8\left( {\frac{{3{r^3} - {r^3}}}{3}} \right)\\ = 8 \times \frac{{2{r^3}}}{3}\\ = \frac{{16{r^3}}}{3}\end{aligned}\)

Therefore, the volume common to two circular cylinders is \(\frac{{16}}{3}{r^3}\).