Question

The density of a semicircular plate with radius a>0 is described by ρ(x,y)=√(x^2+y^2).

The mass of the plate is
Select one:
a. (πa^3)/3


b. πa^2

c. a^3/3


d. (πa^2)/2

Answer 1

The density being $\sqrt{x^2+y^2}$

Let's suppose the we take a small part of the disc at a distance r from center then area of the small part will be $\frac{2\pi r.dr}{2}=\pi rdr$

$x=r\cos\theta\ \&\ y=r\sin\theta$

substituting in the value of density

density = $\sqrt{r^2}=r$

mass = $_0^a\int r.\pi r dr=\pi a^3/3$