Answer to Question #151185 in Calculus for Zeeshan

Question #151185
∫ x² y dx + x y² dy
where d is describe by 0 ≤ x ≤ 1 , 0 ≤ y ≤ x
1
Expert's answer
2020-12-18T10:20:20-0500

Consider the region "D" as shown in the figure below:





The region "D" is a closed smooth curve, so the Green's theorem applicable for this region.


Using Green's theorem, the line integral is evaluated as,


"\\int x^2ydx+xy^2dy"


"=\\iint_D(\\frac{\\partial}{\\partial x}(xy^2)-\\frac{\\partial}{\\partial y}(x^2y))dA"


"=\\iint_D(y^2-x^2)dA"


"=\\int_{x=0}^{1}\\int_{y=0}^{x}(y^2-x^2)dydx"


"=\\int_{x=0}^{1}[\\frac{y^3}{3}-x^2y]_{y=0}^{x}dx"


"=\\int_{x=0}^{1}(\\frac{x^3}{3}-x^3)dx"


"=-\\frac{2}{3}\\int_{x=0}^{1}x^3dx"


"=-\\frac{2}{3}[\\frac{x^4}{4}]_{x=0}^{1}"


"=-\\frac{2}{3}(\\frac{1}{4})"


"=-\\frac{1}{6}"


Therefore, the line integral is "\\int x^2ydx+xy^2dy=-\\frac{1}{6}"

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