Applications of Green's Theorem

Calculus 3 · Axiom Academy

LESSON Applications of Green's Theorem Computing areas using line integrals, and understanding flux across curves. Green's Theorem gives three equivalent ways to compute the area of a region R bounded by a simple closed curve C , traversed counterclockwise . Every one of them turns a double integral (area) into a single line integral along the boundary. Why does this work? Apply Green's Theorem to , , or . In each case the curl is exactly 1 , so is just the area — and Green's Theorem rewrites that as the boundary integral. A planimeter is a mechanical device that measures area by tracing the boundary of a region. As you move the tracer around the curve, a small wheel accumulates the boundary contributions — and when you close the loop, it has computed the enclosed area, no formula required. As you trace the boundary, the wheel measures small contributions (or , or their average). Completing the loop integrates them into the total area — exactly what the formulas above compute. The integral accumulates "horizontal slices," while accumulates "vertical slices." 3. Example: Area of an Ellipse Let's use the symmetric formula to find the area enclosed by the ellipse . Parametrize C counterclockwise: The cross terms combine to the constant ab , so the integral is just times the full sweep — the familiar area of an ellipse. 4. The Flux Form of Green's Theorem Green's Theorem also relates the flux of a vector field across a curve to the divergence inside the region:

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