Question

$\underset{\mathbf{0}}{\overset{\frac{\mathbf{\pi }}{\mathbf{2}}}{\mathbf{\int }}}\sqrt{\mathbf{1}\mathbf{-}\frac{\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\mathbf{\theta }}{\mathbf{9}}\mathbf{d}\mathbf{\theta }}$

Short answer

The value of integral in elliptic form is $E\left(\frac{1}{3}\right)\approx 1.52621$.

Step-by-step solution

Given Information

The integration is $\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\sqrt{1-\frac{{\sin }^2\mathrm{\theta }}{9}\mathrm{d\theta }}$.

Definition of elliptic form.

The elliptic form of the integral is defined as $\mathit{E}\left(\frac{\pi }{2},k\right)\mathbf{=}\underset{\mathbf{0}}{\overset{\frac{\mathbf{\pi }}{\mathbf{2}}}{\mathbf{\int }}}\sqrt{\mathbf{1}\mathbf{-}{\mathbf{k}}^{\mathbf{2}}\mathbf{s}\mathbf{o}{\mathbf{n}}^{\mathbf{2}}\mathbf{\theta }}\mathit{d}\mathit{\theta }$.

Find the value of Integral.

The value of integration is $\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\sqrt{1-\frac{{\sin }^2\mathrm{\theta }}{9}\mathrm{d\theta }}$.

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

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

$$\begin{gathered} l=\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\sqrt{1-\frac{{\sin }^2\theta }{9}}d\theta \\ l=E\left(\frac{\mathrm{\pi }}{2},\frac{1}{3}\right) \\ l\approx 1.52621 \end{gathered}$$

Hence, the value of integral in elliptic form is $E\left(\frac{1}{3}\right)\approx 1.52621$.