Question

For the following regions R and surface densities σ, find the total mass.
(a)R: 0≤y≤sin(pi x/L); 0≤x≤L; σ(x,y) =y.
(b) The region between the semicircles y=sqrt(1-x^2) and y=sqrt(4-x^2) and the segments of the x-axis joining them. The surface density is equal to the distance from any point to the origin.

Answer 1

(a)

$m=\int_{0}^{L}dx\int_{0}^{sin\frac{\pi x}{L}}ydy=\int_{0}^{L}\frac{y^2}{2}\vert_{0}^{sin\frac{\pi x}{L}} dx=\frac{1}{2} \int_{0}^{L} sin^{2} \frac{\pi x}{L} dx=$

$= \int_{0}^{L}(1-cos\frac{2 \pi x}{L}) dx= (x-\frac{L}{2\pi}sin\frac{2 \pi x}{L})|_{0}^{L} =L.$

(b)

$\sigma=r, 0\leq \varphi\leq \pi, 1\leq r\leq 2.$

$m=\int_{0}^{\pi}d\varphi \int_{1}^{2}r\cdot rdr=\pi \int_{1}^{2}r^2dr=\frac13\pi r^3|_{1}^{2}=\frac{7\pi}{3}.$