Question

In Problem 4 to 13, identify each of the integral as an elliptic (see Example 1 and 2). Learn the notation of your computer program (see Problem 3) and then evaluate the integral by computer.

12. $\underset{\mathbf{0}}{\overset{\frac{\mathbf{1}}{\mathbf{2}}}{\mathbf{\int }}}\frac{\sqrt{\mathbf{100}\mathbf{-}{\mathbf{t}}^{\mathbf{2}}}}{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{t}}^{\mathbf{2}}}}\mathit{d}\mathit{t}$

Short answer

The value of integral in elliptic form is $10E\left(\frac{1}{2},\frac{1}{10}\right)\approx 0.52337$.

Step-by-step solution

Given Information

The given value of integration is $\underset{0}{\overset{\frac{1}{2}}{\int }}\frac{\sqrt{100-t^2}}{\sqrt{1-t^2}}dt$.

Definition of elliptic form

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

Find the value of Integral

The value of integration is $\underset{0}{\overset{\frac{1}{2}}{\int }}\frac{\sqrt{100-t^2}}{\sqrt{1-t^2}}dt$.

Factor out 100 the equation becomes as follows:

$\begin{array}{rcl}l& =& \underset{0}{\overset{\frac{1}{2}}{\int }}\frac{\sqrt{100\left(1-{\displaystyle \frac{1}{100}}{t}^2\right)}}{\sqrt{1-t^2}}dt\\ & =& \underset{0}{\overset{\frac{1}{2}}{\int }}\frac{10\sqrt{1-{\displaystyle \frac{1}{100}}{t}^2}}{\sqrt{1-t^2}}dt\end{array}$

The formula for the beta function is $\begin{array}{rcl}E\left(1,K\right)& =& \underset{0}{\overset{1}{\int }}\frac{\sqrt{1-k^2{t}^2}}{\sqrt{1-t^2}}dt\end{array}$.

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

$\begin{array}{rcl}l& =& 10\underset{0}{\overset{\frac{1}{2}}{\int }}\frac{\sqrt{1-{\left({\displaystyle \frac{1}{10}}\right)}^2{t}^2}}{\sqrt{1-t^2}}dt\\ l& =& 10E\left(\frac{1}{2},\frac{1}{10}\right)\\ l& \approx & 0.52337\end{array}$

The value of integral in elliptic form is $\begin{array}{rcl}10E\left(\frac{1}{2},\frac{1}{10}\right)& \approx & 0.52337\end{array}$.