Where C is the union of the segment from (0,0) to (2,0), the segment from (2,0) to (2,4), and the curve y= x^2 from (2,4) to (0,0).
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Since C is a closed curve, let's apply Green's Theorem.
Note that the region R enclosed by C is bounded between y = 0 and y = x^2 for x in [0, 2].
So, ∫c (y^2 dx - x^2 dy)
= ∫∫R [(∂/∂x)(-x^2) - (∂/∂y) y^2] dA
= ∫(x = 0 to 2) ∫(y = 0 to x^2) (-2x - 2y) dy dx
= ∫(x = 0 to 2) (-2xy - y^2) {for y = 0 to x^2} dx
= ∫(x = 0 to 2) (-2x^3 - x^4) dx
= [-(1/2)x^4 - (1/5)x^5] {for x = 0 to 2}
= -72/5.
I hope this helps!
Note that the region R enclosed by C is bounded between y = 0 and y = x^2 for x in [0, 2].
So, ∫c (y^2 dx - x^2 dy)
= ∫∫R [(∂/∂x)(-x^2) - (∂/∂y) y^2] dA
= ∫(x = 0 to 2) ∫(y = 0 to x^2) (-2x - 2y) dy dx
= ∫(x = 0 to 2) (-2xy - y^2) {for y = 0 to x^2} dx
= ∫(x = 0 to 2) (-2x^3 - x^4) dx
= [-(1/2)x^4 - (1/5)x^5] {for x = 0 to 2}
= -72/5.
I hope this helps!