91Ó°ÊÓ

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

Use a graph of $\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{θ}$and the text discussion just before (12.4)to verify the equations (12.4). Note that the area under the$\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{θ}$ graph from $\mathbf{0}\mathbf{-}\frac{\mathbf{π}}{\mathbf{2}}$and the area from $\frac{\mathbf{π}}{\mathbf{2}}\mathbf{-}\mathbf{π}$ are mirror images of each other, and this will be true also for any function of${\mathbf{sin}}^{\mathbf{2}}\mathit{θ}$.

Computer plot graphs of K(k) and E(k)in (12.3)for k = 0 - 1. Also plot 3Dgraphs of $\mathit{F}\left(\mathrm{Ï•},k\right)$and $\mathit{E}\left(\mathrm{Ï•},k\right)$ in (12.1) for k = 0-1and$\mathit{Ï•}$from $\mathbf{0}\mathbf{-}\frac{\mathbf{Ï€}}{\mathbf{2}}$ and also from$\mathbf{0}\mathbf{-}\mathbf{2}\mathbf{Ï€}$. Warning: Be sure you understand the notation used by your computer program; see text discussion just after (12.3) and Example 1.

$\underset{\mathbf{0}}{\overset{\mathbf{1}}{\mathbf{∫}}}\frac{\mathbf{1}}{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{t}}^{\mathbf{2}}}\sqrt{\mathbf{1}\mathbf{-}{\displaystyle \frac{{\mathbf{t}}^{\mathbf{2}}}{\mathbf{4}}}}}\mathit{d}\mathit{t}$

$\underset{\mathbf{0}}{\overset{\frac{\mathbf{π}}{\mathbf{2}}}{\mathbf{∫}}}\sqrt{\mathbf{1}\mathbf{-}\frac{\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\mathbf{θ}}{\mathbf{9}}\mathbf{d}\mathbf{θ}}$

$\underset{\mathbf{0}}{\overset{\frac{\mathbf{π}}{\mathbf{3}}}{\mathbf{∫}}}\frac{\mathbf{1}}{\sqrt{\mathbf{9}\mathbf{-}\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\mathbf{θ}}}\mathit{d}\mathit{θ}\mathbf{.}$

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.

7. $\underset{\mathbf{0}}{\overset{\frac{\mathbf{5}\mathbf{π}}{\mathbf{4}}}{\mathbf{∫}}}\sqrt{\mathbf{25}\mathbf{-}\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\mathbf{θ}}\mathit{d}\mathit{θ}$

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

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.

9. $\underset{\frac{\mathbf{-}\mathbf{1}}{\mathbf{2}}}{\overset{\frac{\mathbf{1}}{\mathbf{2}}}{\mathbf{∫}}}\frac{\mathbf{d}\mathbf{t}}{\sqrt{\mathbf{1}\mathbf{-}{\mathbf{t}}^{\mathbf{2}}}\sqrt{\mathbf{4}\mathbf{-}\mathbf{3}{\mathbf{t}}^{\mathbf{2}}}}$.

x2+y24=1Find the length of arc of the ellipse${\mathit{x}}^{\mathbf{2}}\mathbf{+}\frac{{\mathbf{y}}^{\mathbf{2}}}{\mathbf{4}}\mathbf{=}\mathbf{1}$between$\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{2}\mathbf{)}$and$\mathbf{(}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{,}\sqrt{\mathbf{3}}\mathbf{)}$. (Note that here$\mathit{b}\mathbf{>}\mathit{a}$; see Example 4.).