Evaluating ∫ xy ds over Quarter Circle C

Calculus 3 · Axiom Academy

EXAMPLE Evaluating over a Quarter Circle Parametrize the curve, build the arc-length element ds , and integrate a scalar field along C . Evaluate the line integral , where C is the quarter of the unit circle in the first quadrant, running from (1,0) to (0,1) . The path C : the first-quadrant quarter of the unit circle, traced from (1,0) to (0,1) . Nicely done. You evaluated a scalar line integral by turning the curve into a single parameter and integrating over it. Parametrize: the quarter of the unit circle is for . Arc-length element: , which collapses to ds=dt on the unit circle. Substitute: replace x and y in the integrand with their parametric forms, so . Simplify: the identity makes the integral immediate. The same four moves — parametrize, build ds , substitute, integrate — evaluate any scalar line integral along a curve.

This is the written version of the interactive lesson above. See the full Calculus 3 course.