Finding Mass from Semi Circle and Density

[gucciryan]

Problem Statement:
A thin metal plate is shaped like a semicircle of radius 8 in the right half-plane, centered at the origin. The area density of the metal only depends on x, and is given by ρ(x)=1.4+2.8x kg/m2. Find the total mass of the plate.

I know that we find the area of the semicircle with a radius of 8. So I tried the Integral of (Area of semi circle)*(p(x)) from 0 to 8, but it said it was the wrong answer. Did I do something wrong?

 

[Romsek]

You need to apply the integral that would result in the area had \(\displaystyle \rho=1\), with the variable density \(\displaystyle \rho(x)\)

\(\displaystyle m = \displaystyle \int_{dA}~\rho\)

\(\displaystyle m = 1.4 \displaystyle \int_0^8 \int_{-\sqrt{64-x^2}}^\sqrt{64-x^2}~(1+2x)~dy~dx = 1096.48 ~kg\)