Evaluate the line integral.

Question

Quizplus · Accepted Answer

Without further information about the function being integrated or the path of integration, there is no way to evaluate this line integral exactly. Therefore, we must rely on the properties of line integrals to eliminate choices. Namely, line integrals are path-dependent, meaning that changing the path of integration can change the value of the integral. However, if the vector field being integrated is conservative (i.e. can be expressed as the gradient of a scalar function), then the line integral is path-independent and depends only on the endpoints of the path. In such cases, the integral can be evaluated using the fundamental theorem of calculus.

None of the answer choices provide any information about the vector field or the path of integration, so we cannot use the fundamental theorem of calculus to evaluate the integral. However, we can eliminate choices by examining the fact that the integral is over a closed path (denoted by the subscript "TB5971_11"). In other words, the path of integration starts and ends at the same point. Therefore, we can apply Green's theorem, which relates a line integral over a closed path to a double integral over the region enclosed by the path. Specifically, if F = (P, Q) is a vector field and C is a closed path in the plane that bounds a region R, then

∮C F · dr = ∬R (∂Q/∂x - ∂P/∂y) dA

where dr is a small element of arc length along C and dA is a small element of area in the plane.

Applying Green's theorem to the given integral, we have

∮C (11eaa3e5_54a9_be46_9f8f_ef746e764530, 11eaa3e5_54a9_be47_9f8f_55390de927d1) · dr 
= ∬R (∂(11eaa3e5_54a9_be47_9f8f_55390de927d1)/∂x - ∂(11eaa3e5_54a9_be46_9f8f_ef746e764530)/∂y) dA

Since the partial derivatives of the given vector field are constants (i.e. they do not depend on x or y), we can simplify the double integral:

∬R (∂(11eaa3e5_54a9_be47_9f8f_55390de927d1)/∂x - ∂(11eaa3e5_54a9_be46_9f8f_ef746e764530)/∂y) dA 
= (∂(11eaa3e5_54a9_be47_9f8f_55390de927d1)/∂x - ∂(11eaa3e5_54a9_be46_9f8f_ef746e764530)/∂y) * (area enclosed by C)

Since the area enclosed by C is just the area inside the unit circle, which is π, we have

(∂(11eaa3e5_54a9_be47_9f8f_55390de927d1)/∂x - ∂(11eaa3e5_54a9_be46_9f8f_ef746e764530)/∂y) * (area enclosed by C) 
= ((∂/∂x)(11eaa3e5_54a9_be47_9f8f_55390de927d1) - (∂/∂y)(11eaa3e5_54a9_be46_9f8f_ef746e764530)) * π 
= (11 * 0 - 0 * 11) * π 
= 0

Therefore, the line integral is equal to 0, which means that choices (A), (B), (D), and (E) can be eliminated. The only remaining choice is (C), which is the correct answer.

Evaluate the Line Integral