Thread: Line Integration

$\displaystyle \int _c xy^4ds $ C is the right half of the circle $\displaystyle x^2+y^2=16$

Here is what i did. Since C is the right half of the circle my bounds are from $\displaystyle -\pi/2 \leq t \leq \pi/2 $. I got x = 4cost and y = 4sint. I then take the arclength of that, and multiple that by the original function with the polar coordinates substituted into it which is $\displaystyle \int (4cost)(4sint)^4\sqrt{(-4sint^2)+(4cost^2)}dt $

Im kind of confused on how to simplify the algebra before i integrate. Any step by step guidance would be appreciated .

Originally Posted by purplerain

$\displaystyle \int _c xy^4ds $ C is the right half of the circle $\displaystyle x^2+y^2=16$

Here is what i did. Since C is the right half of the circle my bounds are from $\displaystyle -\pi/2 \leq t \leq \pi/2 $. I got x = 4cost and y = 4sint. I then take the arclength of that, and multiple that by the original function with the polar coordinates substituted into it which is $\displaystyle \int (4cost)(4sint)^4\sqrt{(-4sint^2)+(4cost^2)}dt $

Im kind of confused on how to simplify the algebra before i integrate. Any step by step guidance would be appreciated .

So you have

$\displaystyle 4^6\int_{-\pi/2}^{\pi/2} \cos t \sin^4 t \,dt$ then let $\displaystyle u = \sin t$.

Originally Posted by Danny

So you have

$\displaystyle 4^6\int_{-\pi/2}^{\pi/2} \cos t \sin^4 t \,dt$ then let $\displaystyle u = \sin t$.

I understand the integration, im just confused on how to simplfy the function before i integrate. The book has $\displaystyle 4^5\int costsin^4t 4dt $

I have no idea how they got the 4dt, any help would be appreciated, my algebra is fuzzy.

Originally Posted by purplerain

I understand the integration, im just confused on how to simplfy the function before i integrate. The book has $\displaystyle 4^5\int costsin^4t 4dt $

I have no idea how they got the 4dt, any help would be appreciated, my algebra is fuzzy.

$\displaystyle
x = 4 \cos t, y = 4\sin t \; \text{so}\; dx = -4 \sin t\, dt \; \text{and }\;dy = 4 \cos t\, dt
$
so
$\displaystyle
ds = \sqrt{dx^2+dy^2}\,dt = \sqrt{(-4 \sin t)^2 + (4 \cos t)^2}\, dt$ $\displaystyle = \sqrt{4^2(\sin^2 t+ \cos^2t)}\,dt = \sqrt{4^2}\,dt = 4 dt
$

Originally Posted by Danny

$\displaystyle
x = 4 \cos t, y = 4\sin t \; \text{so}\; dx = -4 \sin t\, dt \; \text{and }\;dy = 4 \cos t\, dt
$
so
$\displaystyle
ds = \sqrt{dx^2+dy^2}\,dt = \sqrt{(-4 \sin t)^2 + (4 \cos t)^2}\, dt$ $\displaystyle = \sqrt{4^2(\sin^2 t+ \cos^2t)}\,dt = \sqrt{4^2}\,dt = 4 dt
$

Thank you!