Use Green's Theorem to evaluate where C is curve bounded by and oriented clockwise.
D) 0

Question

Use Green's Theorem to evaluate  where C is curve bounded by  and  oriented clockwise.
D) 0

Quizplus · Accepted Answer

Green's Theorem states that 
∫∫(∂Q/∂x - ∂P/∂y)dA = ∫(Pdx + Qdy) 
where C is a positively oriented, piecewise-smooth boundary of a simply connected, bounded region D in the xy-plane, P and Q have continuous partial derivatives in an open region that contains D, and the curve is traversed once in the counterclockwise direction.

In this problem, the curve is oriented clockwise, so we need to reverse the signs of P and Q. Let P = 5x^2y, and Q = x^3. Then, ∂Q/∂x = 3x^2 and ∂P/∂y = 5x^2, so

∫∫(∂Q/∂x - ∂P/∂y)dA = ∫∫(3x^2 - 5x^2)dA = -2∫∫x^2 dA

We can use polar coordinates to evaluate this double integral, noting that the region D is a disk with radius 2 centered at (0,0):

-2∫∫x^2 dA = -2∫θ=0→2π ∫r=0→2 (r^3 cos^2 θ)drdθ

= -2∫θ=0→2π(cos^2 θ) ∫r=0→2 (r^3)drdθ

= -16π/5

Finally, we have

∫(Pdx + Qdy) = ∫-C (5x^2y dy - x^3 dx)

= ∫C (5x^2y dy - x^3 dx)

= -(-16π/5) = 16π/5

Since the answer is nonzero, we can eliminate choices A, B, and C. Therefore, the correct answer is D.

Use Green's Theorem to Evaluate Where C Is Curve