Solve ∫ (from 0 to 2 pi) of int_0^1sqrt{2}rdrdθ

$$\int _ { 0 } ^ { 2 \pi } \int _ { 0 } ^ { 1 } \sqrt { 2 } r d r d \theta$$

Evaluate

$$\pi \sqrt{2}\approx 4.442882938$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \int _{0}^{1}\sqrt{2}r\mathrm{d}r\mathrm{d}\theta$$

Find the integral of \int _{0}^{1}\sqrt{2}r\mathrm{d}r using the table of common integrals rule \int a\mathrm{d}\theta =a\theta .

$$\int _{0}^{1}\sqrt{2}r\mathrm{d}r\theta$$

Simplify.

$$\frac{\sqrt{2}\theta }{2}$$

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.

$$\frac{1}{2}\times 2^{\frac{1}{2}}\times 2\pi -\frac{1}{2}\times 2^{\frac{1}{2}}\times 0$$

Simplify.

$$\sqrt{2}\pi$$