Q 39. - Calculus

Question

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

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

Step-by-Step Solution

Answer

The mass of $R$ is $m=\frac{1}{2}kh{b}^2$.

Step 1: Given Information

The vertices of rectangular region are $(0,0),(b,0),(0,h)\text{ and }(b,h)$

$\rho (x,y)=kx$

Step 2: Calculation of Mass

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

$m={\int }_0^b{\int }_0^{h}kxdydx$ as $\rho (x,y)=kx$

Solving inner integral first $m={\int }_0^{b}kx[y{]}_0^{h}dx$

$m={\int }_0^{b}kxhdx$

$m=kh{\int }_0^{b}xdx$

$m=kh{\left[\frac{x^2}{2}\right]}_0^b$

$m=\frac{1}{2}kh{b}^2$

The required mass is $m=\frac{1}{2}kh{b}^2$