# Line Integral

#### Em Yeu Anh

Evaluate $$\displaystyle \int_Csinxdx+cosydy$$ where C is the top half of the circle $$\displaystyle x^2+y^2=4$$ from (2,0) to (-2,0) and the line segment from (-2,0) to (-3,2).

Haven't done vector calc in awhile, still a little bit rusty, can someone assist me with getting the parametric equations for the line segment?

#### Bruno J.

MHF Hall of Honor
The line segment from $$\displaystyle p_1$$ to $$\displaystyle p_2$$ can be parametrized as $$\displaystyle r(t)=(1-t)p_1+tp_2$$, $$\displaystyle 0 \leq t \leq 1$$.

• Em Yeu Anh

#### 11rdc11

Evaluate $$\displaystyle \int_Csinxdx+cosydy$$ where C is the top half of the circle $$\displaystyle x^2+y^2=4$$ from (2,0) to (-2,0) and the line segment from (-2,0) to (-3,2).

Haven't done vector calc in awhile, still a little bit rusty, can someone assist me with getting the parametric equations for the line segment?

For the circle let

$$\displaystyle x=2\cos{t}$$ and $$\displaystyle dx = -2\sin{t}$$

$$\displaystyle y =2\sin{t}$$ and $$\displaystyle dy = 2\cos{t}$$

Now sub these values for

$$\displaystyle \int_Csinxdx+cosydy$$

and you get

$$\displaystyle C_1 = \int^{\pi}_{0} \bigg(-2\sin{t}\sin{(2\cos{t})}+ 2\cos{t}\cos{(2\sin{t})}\bigg)dt$$

For $$\displaystyle C_2$$

$$\displaystyle x = -2 -t$$ and $$\displaystyle dx= -1$$

$$\displaystyle y =2t$$ and $$\displaystyle dy = 2$$

and then just sub in like before.

• Em Yeu Anh