Using polar coordinates, ∬ r dr dθ over the full disk of radius 2 equals 2π × ∫ 0² r dr. Evaluate the area of the disk. Set the slider to your answer.

Physics I / Change of Variables and the Jacobian

Practice question · Estimate

Using polar coordinates, $\iint r\,dr\,d\theta$ over the full disk of radius 2 equals $2\pi \times \int_0^2 r\,dr$ . Evaluate the area of the disk. Set the slider to your answer.

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Hints

$\int_0^2 r\,dr = r^{2}/2 \big|_0^2 = 2$ .
Multiply by the full angle $2\pi$ .

Show the answer

12.566 (answers within ±0.4 count)

Why

$2\pi \times 2 = 4\pi \approx 12.57$ , reproducing $\pi R^{2}$ from first principles. The Jacobian $r$ is exactly what makes the polar computation land on the true area.

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Using polar coordinates, ∬r dr dθ\iint r\,dr\,d\theta∬rdrdθ over the full disk of radius 2 equals 2π×∫02r dr2\pi \times \int_0^2 r\,dr2π×∫02​rdr. Evaluate the area of the disk. Set the slider to your answer.

Practise Change of Variables and the Jacobian

More questions on Change of Variables and the Jacobian

Using polar coordinates, $\iint r\,dr\,d\theta$ over the full disk of radius 2 equals $2\pi \times \int_0^2 r\,dr$ . Evaluate the area of the disk. Set the slider to your answer.