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MathCalculus

Evaluate the Line Integral Along the Curve C: y² = x³ with (y − x) dx + x²y dy

Parameterize the directed cusp y²=x³ and evaluate the differential-form line integral from (1,-1) to (1,1).

Question

Evaluate the line integral along the curve CC, where C:y2=x3C:y^2=x^3 is traversed from (1,−1)(1,-1) to (1,1)(1,1):

∫C(y−x) dx+x2y dy.\int_C (y-x)\,dx+x^2y\,dy.

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Evidence boundary

The result uses the directed cusp path in Occidental question 11 and a differential-form line integral with dx and dy terms. It is not an arc-length integral; reversing the path reverses its sign.