Question

Let $R$be rectangular region of vertices $(0,0),(b,0),(0,h),\text{and}(b,h)$

If the density at each point in $R$is proportional to the square of the point’s distance from the $y$-axis, find the mass of$R$.

Short answer

The mass is$m=\frac{1}{3}kh{b}^3$.

Step-by-step solution

Given Information

Vertices of rectangular region is $(0,0),(b,0),(0,h)\text{and}(b,h)$

Calculation of Mass

The mass is calculated as $m={\iint }_{\Omega }\rho (x,y)dA$

and $\rho (x,y)=k{x}^2$

$m={\int }_0^b{\int }_0^{h}k{x}^{2}dydx$

$m={\int }_0^{b}k{x}^2[y{]}_0^{h}dx$

$m={\int }_0^{b}k{x}^{2}hdx=kh{\int }_0^b{x}^{2}dx$

$\Rightarrow m=kh{\left[\frac{x^3}{3}\right]}_0^b$

$m=\frac{1}{3}kh{b}^3$

The mass is$m=\frac{1}{3}kh{b}^3$