Question

Find the circumference of the ellipse$\mathbf{4}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{9}{\mathit{y}}^{\mathbf{2}}\mathbf{=}\mathbf{36}$.

Short answer

The circumference of ellipse is $12E\left(\frac{\mathrm{\pi }}{2},\frac{\sqrt{5}}{3}\right)\approx 1.32211$.

Step-by-step solution

Given Information

The equation of ellipse is $4x^2+9y^2=36$.

Definition of the circumference of ellipse.

The circumference of ellipse defined as $\mathbf{4}\mathit{a}\underset{\mathbf{0}}{\overset{\frac{\mathbf{\pi }}{\mathbf{2}}}{\mathbf{\int }}}\sqrt{\mathbf{1}\mathbf{-}\frac{{\mathbf{a}}^{\mathbf{2}}\mathbf{-}{\mathbf{b}}^{\mathbf{2}}}{{\mathbf{a}}^{\mathbf{2}}}\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\mathbf{\theta }}\mathit{d}\mathit{\theta }$.

Find the value of Integral.

The equation of ellipse is $4x^2+9y^2=36$.

Substitute the values given below in the equation mentioned above.

a² = 9

b² = 4

The circumference of ellipse defined as $C=4a\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\sqrt{1-\frac{{\mathrm{a}}^2-{\mathrm{b}}^2}{{\mathrm{a}}^2}{\sin }^2\mathrm{\theta }}d\theta$.

The equation becomes as follows.

$$\begin{gathered} C=4a\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\sqrt{1-\frac{{\mathrm{a}}^2-{\mathrm{b}}^2}{{\mathrm{a}}^2}{\sin }^2\mathrm{\theta }}d\theta \\ C=4\times 2\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\sqrt{1-\frac{9-4}{9}{\sin }^2\mathrm{\theta }}d\theta \\ C=4\times 2\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\sqrt{1-\frac{5}{4}{\sin }^2\mathrm{\theta }}d\theta \end{gathered}$$

The formula for the beta function is $E\left(\frac{\mathrm{\pi }}{2},K\right)=\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\sqrt{1-k^2{\sin }^2\theta }d\theta$.

Equate the above equation with the value of C, the value of C becomes follows.

$$\begin{gathered} C=4\times 2\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\sqrt{1-\frac{5}{4}{\sin }^2\mathrm{\theta }}d\theta \\ C=12E\left(\frac{\mathrm{\pi }}{2},\frac{\sqrt{5}}{3}\right) \\ C\approx 1.32211 \end{gathered}$$

Hence, the solution is mentioned below.

The circumference of ellipse is $12E\left(\frac{\mathrm{\pi }}{2},\frac{\sqrt{5}}{3}\right)\approx 1.32211$.