Question

Use a double integral to derive the area of the region between circles of radius a and b with
α
≤
θ
≤
β
α≤θ≤β
. See the image below for a sketch of the region.

Answer 1

$$S=\iint\limits_{\begin{matrix} a^2\le x^2+y^2\le b^2\\ \alpha\le\theta\le\beta \end{matrix}}1dxdy =\left[\begin{array}{l} x=r\cdot\cos\theta\\ y=r\cdot\sin\theta\\ dxdy=rdrd\theta\\ r\in[a;b]\\ \alpha\le\theta\le\beta \end{array}\right]=\\[0.3cm] =\int\limits_\alpha^\beta d\theta\cdot\left(\int\limits_a^brdr\right)=\left.\theta\right|_\alpha^\beta\cdot\left(\left.\frac{r^2}{2}\right|_a^b\right)=\left(\beta-\alpha\right)\cdot\frac{b^2-a^2}{2}$$

Conclusion,

$$\boxed{S=\frac{\left(\beta-\alpha\right)\cdot\left(b^2-a^2\right)}{2}}$$