Using FTC for ∫ F·dr over Curve C where F = <2xy, x²+1>

Calculus 3 · Axiom Academy

EXAMPLE Using the FTC for with Evaluate a line integral by finding a potential function — no parametrization needed. Evaluate , where and C is any smooth curve running from the point A=(1,0) to the point B=(2,3) . Both routes share the endpoints A=(1,0) and B=(2,3) . Because is conservative, the integral is the same along either — it depends only on the endpoints. Nice work! You evaluated a line integral without ever touching a parametrization. Here's the idea to carry forward: Conservative test: is conservative on a simply-connected region when . Build the potential: integrate P in x , then match to recover the y -only piece g(y) . FTC shortcut: once , — here 15-0=15 . Path independence: the value depends only on the endpoints, so any curve from A to B gives the same answer. For a conservative field, a messy line integral collapses into a single subtraction — that's the power of the Fundamental Theorem for Line Integrals.

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