Evaluate the line integral, where c is the given curve. ∫c sin(x) dx + cos(y) dy, where c consists of the top half of the circle x^2 + y^2 = 4 from (2, 0) to (−2, 0) and the line segment from (−2, 0) to (−3, 2)

Step-by-step explanation:

Previous concepts

Fundamental Theorem of Line Integrals and into Green’s Theorem

Suppose a Curve C given by the vector function with a= r(a) and b= r(b). Then we have this:

Where Vf represnent the gradient of the function f.

Solution to the problem

We want to find this integral:

And the region C is given by the top half circle with equation with radius 2 and we have a line segment from (-2,0) to (-3,2).

In order to solve this problem we can use this form:

Where is a bivariate function.

For our case our function F is given by: and that represent our vector field.

We can find the potential function or the antiderivate like this:

At the point (2,0) we can find the potential function and we got:

At the point (-3,2) we can find the potential function and we got:

And then the integral is given by:

(a) the volume of the cone is 20/3 in3

hence, option a is correct.