Use Green's Theorem to evaluate the line integral along the given positively oriented curve.
∫_C (8y + 2e^(∑x)) dx + (11x + 8 cos(y^2)) dy
C is the boundary of the region enclosed by the parabolas y = x^2 and x = y^2.
Step 1
We note that C is a positively-oriented, smooth, simple closed curve. Green's Theorem tells us that in this situation, if D is the region bounded by C, then
∫_C P dx + Q dy = ∬_D (∂Q/∂x - ∂P/∂y) dA.
Step 2
Therefore, by Green's Theorem we have the following.
∫_C (8y + 2e^(∑x)) dx + (11x + 8 cos(y^2)) dy = ∬_D (∂/∂x(11x + 8 cos(y^2)) - ∂/∂y(8y + 2e^(∑x))) dA
= ∬_D (11 - 8) dA
= ∬_D 3 dA
Step 3
Since D is the region enclosed by y = x^2 and x = y^2, then
3 ∬_D dA = 3 ∫_0^1 ∫_{y^2}^{∑y} dx dy.
Step 4
We have
∫_{y^2}^{∑y} dx = 1.