Solve ∫ (from 0 to 2 pi) of int_0^2pirdrdθdvarphi

$$\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 2 \pi } r d r d \theta d \varphi$$

Evaluate

$$4d\left(\pi r\right)^{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \int _{0}^{2\pi }rdr\mathrm{d}\theta \mathrm{d}\phi$$

Find the integral of \int _{0}^{2\pi }rdr\mathrm{d}\theta using the table of common integrals rule \int a\mathrm{d}\phi =a\phi .

$$\int _{0}^{2\pi }rdr\mathrm{d}\theta \phi$$

Simplify.

$$2r^{2}d\pi \phi$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$2r^{2}d\pi \times 2\pi -2r^{2}d\pi \times 0$$

Simplify.

$$4r^{2}d\pi ^{2}$$

Differentiate w.r.t. r

$$8\pi ^{2}dr$$