Question

To calculate the volume of the described solid (an elliptical region).

Short answer

The volume of an elliptical region is \(24\).

Step-by-step solution

Given data

The boundary curve of an elliptical region is \(9{x^2} + 4{y^2} = 36\) …..(1)

Cross-sections perpendicular to the \(x\)-axis are isosceles right triangles with hypotenuse in the base.

Concept used of Volume

Volumeis a scalar quantity expressed in the amount of three-dimensional space enclosed by a closed surface.

Solve to find the volume

Consider the side length of the isosceles right triangle is\(l\)and the hypotenuse is 2 y.

Rearrange Equation (1).

\(\begin{aligned}{}9{x^2} + 4{y^2} = 36\\4{y^2} = 36 - 9{x^2}\\2y = \sqrt {36 - 9{x^2}} \\y = \frac{3}{2}\sqrt {4 - {x^2}} \end{aligned}\)

Show the side length of the isosceles right triangle.

\(\begin{aligned}{}{l^2} + {l^2} = {(2y)^2}\\2{l^2} = 4{y^2}\\{l^2} = 2{y^2}\end{aligned}\)

The cross-section of an elliptical region as shown in Figure 1.

The region lies between \(a = - 2\) and \(b = 2\).

The expression to find the volume of an elliptical region as shown below.

\(V = \int_a^b A (x)dx\) …..(2)

Find the area of an isosceles right triangle as shown below.

\(\begin{aligned}{}A(x) = \frac{1}{2} \times l \times l\\ = \frac{1}{2}{l^2}\\ = \frac{1}{2} \times 2{y^2}\\ = {y^2}\end{aligned}\)

Simplify further,

\(\begin{aligned}{}A(x) = {\left( {\frac{3}{2}\sqrt {4 - {x^2}} } \right)^2}\\ = \frac{9}{4}\left( {4 - {x^2}} \right)\end{aligned}\)

Substitute \( - 2\) for a, 2 for \(b\), and \(\frac{9}{4}\left( {4 - {x^2}} \right)\) for \(A(x)\) in Equation (2).

\(\begin{aligned}{}V = \int_{ - 2}^2 {\frac{9}{4}} \left( {4 - {x^2}} \right)dx\\ = \frac{9}{2}\int_0^2 {\left( {4 - {x^2}} \right)} dx\\ = \frac{9}{2}\left( {4x - \frac{{{x^3}}}{3}} \right)_0^2\\ = \frac{9}{2}\left( {4 \times 2 - \frac{{{2^3}}}{3} - 0} \right)\end{aligned}\)

Simplify further,

\(\begin{aligned}{}V = \frac{9}{2} \times \frac{{24 - 8}}{3}\\ = 3 \times 8\\ = 24\end{aligned}\)

Therefore, the volume of an elliptical region is 24 .