Curvilinear integral

Calculus Level 4

Calculate the line(curvilinear) integral ∮ γ x y d x + ( x 2 − y 2 ) d y \oint_{\gamma} xy \space dx + (x^2 - y^2) \space dy along the path γ \gamma (AOB) of the parabola y 2 = x y^2 = x , where A = ( 1 , − 1 ) , O = ( 0 , 0 ) , B = ( 1 , 1 ) A = (1, -1), O = (0,0), B = (1, 1) .

The above integral has a closed form. Enter your answer to 3 decimal places


The answer is 0.5333333333333.

Guillermo Templado by Guillermo Templado , 46, Spain

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1 solution

We can suppose that the parametric representation of γ ( t ) = ( t 2 , t ) \gamma(t) = (t^2, t) with the canonical orientation, in this case clockwise. Then ∮ γ x y d x + ( x 2 − y 2 ) d y = ∫ − 1 1 2 t 4 + t 4 − t 2 d t = ∫ − 1 1 3 t 4 − t 2 d t = 8 15 ≈ 0.5333333 \oint_{\gamma} xy \space dx + (x^2 - y^2) dy = \displaystyle \int_{-1}^{1} 2t^4 + t^4 - t^2 \space dt = \int_{-1}^{1} 3t^4 - t^2 \space dt = \frac{8}{15} \approx \boxed{0.5333333}

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