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Evaluate a line integral

Short answer

Parameterize the parabola as y = x^2, dy = 2x dx, x from 0 to 1. The line integral becomes ∫_0^1 (x^2 - 3x^4) dx = -4/15.

Evaluate the line integral ∫_C [(x² + y²) i − 2xy j]·dr, where C is the parabola x² = y from (0,0) to (1,1).

Let F = (x² + y²) i − 2xy j and dr = dx i + dy j + dz k, so

F·dr = (x² + y²) dx − 2xy dy

Along C: y = x², dy = 2x dx, with x from 0 to 1.

∫_C F·dr = ∫₀¹ [(x² + x⁴) − 2x(x²)(2x)] dx
         = ∫₀¹ (x² + x⁴ − 4x⁴) dx
         = ∫₀¹ (x² − 3x⁴) dx
         = [x³/3 − 3x⁵/5]₀¹
         = 1/3 − 3/5
         = (5 − 9)/15
         = −4/15

Answer: ∫_C F·dr = −4/15.

✓ Answer checked by Claude Opus 5.5 (high effort)

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