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.

8. $\underset{\mathbf{0}}{\overset{\frac{\sqrt{\mathbf{3}}}{\mathbf{2}}}{\mathbf{\int }}}\frac{\sqrt{\mathbf{49}\mathbf{-}\mathbf{4}{\mathbf{t}}^{\mathbf{2}}}}{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{t}}^{\mathbf{2}}}}\mathit{d}\mathit{t}$

Short answer

The value of integral in elliptic form is $7E\left(\frac{\sqrt{3}}{2},\frac{2}{7}\right)\approx 6.11662$.

Step-by-step solution

Given Information

The value of integration is $\underset{0}{\overset{\frac{\sqrt{3}}{2}}{\int }}\frac{\sqrt{49-4{\mathrm{t}}^2}}{\sqrt{1-{\mathrm{t}}^2}}dt$.

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 $\underset{0}{\overset{\frac{\sqrt{3}}{2}}{\int }}\frac{\sqrt{49-4t^2}}{\sqrt{1-t^2}}dt$.

Factor out 49 the equation becomes as follows,

$\begin{array}{rcl}l& =& \underset{0}{\overset{\frac{\sqrt{3}}{2}}{\int }}\frac{\sqrt{49\left(1-{\displaystyle \frac{4t^2}{49}}\right)}}{\sqrt{1-t^2}}dt\\ & =& 7\underset{0}{\overset{\frac{\sqrt{3}}{2}}{\int }}\frac{\sqrt{\left(1-{\displaystyle \frac{4t^2}{49}}\right)}}{\sqrt{1-t^2}}dt\end{array}$

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, the value of I becomes follows.

$$\begin{gathered} l=7\underset{0}{\overset{\frac{\sqrt{3}}{2}}{\int }}\frac{\sqrt{1+{\displaystyle \frac{2^2{t}^2}{7^2}}}}{\sqrt{1-t^2}}dt \\ l=7E\left(\frac{\sqrt{3}}{2},\frac{2}{7}\right) \\ l\approx 6.11662 \end{gathered}$$

Hence, the solution is mentioned below.

The value of integral in elliptic form is $\begin{array}{rcl}7E\left(\frac{\sqrt{3}}{2},\frac{2}{7}\right)& \approx & 6.11662\end{array}$.