Up 244.14.4.26: Plotting $r^7$ Integrals

[karush]

$\tiny{up 244.14.4.26}$
$\textit{a. Sketch the region of Intregration}$
$\textit{b. convert the sum of integrals to a Cartesian sum of Integrals:}$
\begin{align*}\displaystyle
I_{26}&=\int_{0}^{\tan^{-1}(4/3)}
\int_{0}^{3\sec{\theta}}
r^7 \, dr \, d\theta
+\int_{\tan^{-1}(4/3)}^{\pi/2}
\int_{0}^{4\csc\theta}
r^7 \, dr \, d\theta\\
&=\int_{0}^{\tan^{-1}(4/3)}\Biggr|\frac{r^8}{8}\Biggr|_0^{3\sec{\theta}} \, d\theta
+\int_{\tan^{-1}(4/3)}^{\pi/2}\Biggr|\frac{r^8}{8}\Biggr|_{0}^{4\csc\theta} \, d\theta
\end{align*}
next ?

OK first I don't how you plot $r^7$ on Desmos

 

[tkhunny]

$\tiny{up 244.14.4.26}$
$\textit{a. Sketch the region of Intregration and convert the sum of integrals to a Cartesian sum of Integrals:}$
\begin{align*}\displaystyle
I_{26}&=\int_{0}^{\tan^{-1}(4/3)}
\int_{0}^{3\sec{\theta}}
r^7 \, dr \, d\theta
+\int_{\tan^{-1}(4/3)}^{\pi/2}
\int_{0}^{4\csc\theta}
r^7 \, dr \, d\theta\\
&=\int_{0}^{\tan^{-1}(4/3)}\Biggr|\frac{r^8}{8}\Biggr|_0^{3\sec{\theta}} \, d\theta
+\int_{\tan^{-1}(4/3)}^{\pi/2}\Biggr|\frac{r^8}{8}\Biggr|_{0}^{4\csc\theta} \, d\theta
\end{align*}
next ?

OK first I don't how you plot $r^7$ on Desmos

Can you plot:

1) $$\theta = \tan^{-1}(4/3)$$

2) $$r = 3\cdot\sec(\theta)$$

3) $$\theta = \pi/2$$

4) $$r = 4\cdot\csc(\theta)$$

Why are you trying to plot $r^7$? $r^6$ may be more appropriate. Why?

 

[karush]

well I tried but??View attachment 7693

 

[tkhunny]

Interesting. Why did you choose to abandon $\theta$ and insert y and x on $\theta = \pi/2$ and the inverse tangent?

$\theta = \pi/2$ is the y-axis.

Give some thought to the inverse tangent, too.

You could just substitute some version of the basic identities and rearrange a little.

$y = r\cdot\sin(\theta)$

$x = r\cdot\cos(\theta)$

$r^{2} = x^{2}+y^{2}$

$\theta = $ ?? I'll let you track down this one. :-)

 

[karush]

$\theta = \tan^{-1}(4/3)$ doesn't plot
I assume we are finding the area of a wedge?
\begin{align*}
&=\int_{0}^{\tan^{-1}(4/3)}\Biggr|\frac{r^8}{8}\Biggr|_0^{3\sec{\theta}} \, d\theta
+\int_{\tan^{-1}(4/3)}^{\pi/2}\Biggr|\frac{r^8}{8}\Biggr|_{0}^{4\csc\theta} \, d\theta \\
&=3\int_{0}^{\tan^{-1}(4/3)} r^7 \sec{\theta}\, dr
\, +
4\int_{\tan^{-1}(4/3)}^{\pi/2} r^7 \csc(\theta) dr
\end{align*}
so far

 

[tkhunny]

Are you sure you're understanding the problem statement? I don't see any instruction to EVALUATE the integrals. Just draw the pictures and transform to Cartesian Coordinates.

 

[karush]

I don't don't think my drawing is correct?
or is it?

 

[tkhunny]

I don't don't think my drawing is correct?
or is it?

You haven't fixed $\theta = \pi/2$

 

[karush]

You haven't fixed $\theta = \pi/2$

that expression won't plot in desmos

is the same as $x=\frac{\pi}{2}$?

 

[tkhunny]

that expression won't plot in desmos

is the same as $x=\frac{\pi}{2}$?

Absolutely not.