Q 42. - Calculus

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 $R$.

Step-by-Step Solution

Answer

The mass is $m=\frac{1}{3}kh{b}^3$.

Step 1: Given Information

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

Step 2: 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 $m=\frac{1}{3}kh{b}^3$