Double Integrals in Polar

Calculus 3 · Axiom Academy

LESSON Double Integrals in Polar Coordinates The area element becomes — and that Jacobian factor r is the make-or-break detail. Polar coordinates pin down a point with two numbers: its distance from the origin, , and the angle it makes with the positive x -axis. A right triangle bridges the two coordinate systems. 2. The Area Element Grows With r Here is the one detail this whole lesson turns on. A polar grid cell spanning a small and is a curved rectangle : its radial width is , but its arc length is — and that arc stretches with r . So a cell far from the pole covers more area than one near it. Polar coordinates pay off when the region has circular edges. Three show up constantly — and in each one the bounds are constants , which is exactly what keeps the limits from fighting you. 4. Setting Up — and Why the r Changes the Answer Put the pieces together. Replace x and y with and , attach the Jacobian r , and integrate r on the inside, on the outside: Worked example — the area of a disk Take f = 1 over a disk of radius R . The integral should return the disk's own area — and, with the r , it does: You have seen why a polar double integral carries a factor of r — and exactly what goes wrong without it. Scroll up to revisit any step.

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