Evaluating a Line Integral

[jenn9580]

Evaluate the line integral, where C is the given curve.

C [sup:26wi2ldy]sin(x)dx + cos(y)dy[/sup:26wi2ldy]

C consists of the top half of the circle x^2+y^2=25 from (5, 0) to (-5, 0) and the line segment from (-5, 0) to (-6, 2).

I know how to work this type of problem but I am not sure of how to start it. The parametrization of the circle is
x=5cos(t)
y=5sin(t)
with t from 0 to pi.

How do I go about plugging back in?

 

[daon]

First find f(x), g(y), dy and dx with respect to t. Then you will have

\(\displaystyle \int_C f(x)dx + g(y)dy = \int \left (r(t)dt + s(t)dt \right ) = \int_{t_0}^{t_1} (r+s)(t) dt\)

 

[jenn9580]

It's still not clicking.

f(x)=sin(x)
g(y)=cos(y)

parametrizations are:
x=5cos(t) dx=-5sin(t)dt
y=5sin(t) dy= 5cos(t)dt

Where do I go from here?

 

[daon]

how about rewriting the integral without any x's and y's?

edit: You may want to use:

\(\displaystyle x=t\)
\(\displaystyle y=\sqrt{25-t^2}\)