91Ó°ÊÓ

The equation of ellipse is $x^2+\frac{y^2}{4}=1$.

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

The equation of ellipse is $x^2+\frac{y^2}{4}=1$.

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

Substitute the values given below in the equation mentioned above.

role="math" localid="1664304134132" $$\begin{gathered} a^2=1 \\ {b}^2=4 \end{gathered}$$

$$\begin{gathered} C=4a\underset{0}{\overset{\frac{\mathrm{π}}{2}}{∫}}\sqrt{1-\frac{a^2-b^2}{a^2}{\sin }^2\mathrm{θ}}d\mathrm{θ} \\ {C}_1=2\underset{0}{\overset{\frac{\mathrm{π}}{2}}{∫}}\sqrt{1-\frac{-1+4}{1}{\sin }^2\mathrm{θ}}d\mathrm{θ}-2\underset{0}{\overset{\mathrm{ϕ}}{∫}}\sqrt{1-\frac{-1+4}{1}{\sin }^2\mathrm{θ}}d\mathrm{θ} \\ C=2\underset{0}{\overset{\frac{\mathrm{π}}{2}}{∫}}\sqrt{1-\frac{3}{4}{\sin }^2\mathrm{θ}}d\mathrm{θ}-2\underset{0}{\overset{\mathrm{ϕ}}{∫}}\sqrt{1-\frac{3}{4}{\sin }^2\mathrm{θ}}d\mathrm{θ} \end{gathered}$$

The equation becomes as follows and the limits are mentioned in the formula.

The formula for the beta function is $E(\frac{\mathrm{π}}{2},K)=\underset{0}{\overset{\frac{\mathrm{π}}{2}}{∫}}\sqrt{1-k^2{\sin }^2\mathrm{θ}}d\mathrm{θ}$.

Equate the above equation with the value of C₁, the value of C₁ becomes follows.

$$\begin{gathered} C_1=2\underset{0}{\overset{\frac{\mathrm{π}}{2}}{∫}}\sqrt{1-\frac{3}{4}{\sin }^2\mathrm{θ}}d\mathrm{θ}-2\underset{0}{\overset{\mathrm{ϕ}}{∫}}\sqrt{1-\frac{3}{4}{\sin }^2\mathrm{θ}}d\mathrm{θ} \\ {C}_1=2E(\frac{\mathrm{π}}{2},\frac{\sqrt{3}}{2})-2E(\mathrm{ϕ},\frac{\sqrt{3}}{2}) \\ {C}_1≈0.29161 \end{gathered}$$

Hence, the solution is mentioned below.

The length of arc of ellipse is $2E(\frac{\mathrm{π}}{2},\frac{\sqrt{3}}{2})-2E(\mathrm{ϕ},\frac{\sqrt{3}}{2})≈0.29161$.