# Thread: Find the work done by F in moving a body counterclockwise around a curve

1. ## Find the work done by F in moving a body counterclockwise around a curve

$\displaystyle F=(x^{2}+y^{3})i+(2xy)j$ $\displaystyle C:x^{2}+y^{2}=9$

I'm not totally sure how to do this problem. So far, I've done this:

$\displaystyle r(t)=rcos(t)i+rsin(t)j$ => $\displaystyle 3cos(t)i+3sin(t)j$ $\displaystyle 0\leq t \leq 2\pi$

$\displaystyle F=(9cos^{2}(t)+27sin^{3}(t))i+18sin(t)cos(t)j$

$\displaystyle \frac{dr}{dt}=-3sin(t)i+3cos(t)j$

$\displaystyle F * \frac{dr}{dt}=$=$\displaystyle -3sin(t)(9cos^{2}(t)+27sin^{3}(t))+3cos(t)(18sin(t) cos(t))$

=> $\displaystyle -27sin(t)cos^{2}(t)+81sin^{4}+48sin(t)cos^{2}(t)$

$\displaystyle \displaystyle \int^{2\pi}_{0}81sin^{4}+21sin(t)cos^{2}(t)dt$

From here, I don't know how to calculate the integral. Is there a trig identity that simplifies the problem? Do I have to integral by parts or substitution? I plugged it into wolframalpa and got a solution but it didn't give me what I really wanted, the steps on how to solve the integral.

Am I even approaching the problem correctly? Thanks

2. Originally Posted by downthesun01
$\displaystyle F=(x^{2}+y^{3})i+(2xy)j$ $\displaystyle C:x^{2}+y^{2}=9$
$\displaystyle \displaystyle \int^{2\pi}_{0}81sin^{4}+21sin(t)cos^{2}(t)dt$ From here, I don't know how to calculate the integral.
For the first one: $\displaystyle \sin ^4t=[(1/2)(1-\cos 2t]^2=\ldots$. For the second one: $\displaystyle v=\cos t$.

Am I even approaching the problem correctly? Thanks
Yes, you are.

Regards.

3. Thanks for the help. I'm not getting the tips for integration that you gave. For the first one, is that a trig substitution? And for the second one, am I supposed to used integration by parts or is that a substitution? I'm really terrible at integrating trig functions.

Thank you, I've computed the integrals now.

4. Originally Posted by downthesun01
For the first one, is that a trig substitution?
No, it isn't. We use trigonometric formulas.

And for the second one, am I supposed to used integration by parts or is that a substitution? I'm really terrible at integrating trig functions.
It is a substitution:

$\displaystyle \int \sin t\cos^2tdt=-\int v^2dv=-v^3/3+C=-\cos^3t/3+C$

Regards.