Q 38. - Calculus

Question

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

Find the centroid of $R$

Step-by-Step Solution

Answer

The centroid is $\overline{x}=\frac{b}{2},\overline{y}=\frac{h}{2}$

Step 1: Given Information

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

Step 2: Calculation of x -

Centroid is expressed as $\overline{x}=\frac{{\iint }_{\Omega }x\rho (x,y)dA}{{\iint }_{\Omega }\rho (x,y)dA}\text{ and }\overline{y}=\frac{{\iint }_{\Omega }y\rho (x,y)dA}{{\iint }_{\Omega }\rho (x,y)dA}$

Density is uniform, $\rho (x,y)=1$

$\overline{x}=\frac{{\int }_0^b{\int }_0^{h}xdydx}{{\int }_0^b{\int }_0^{h}dydx}$

Solving inner integral first

$\overline{x}=\frac{{\int }_0^{b}x[y{]}_0^{h}dx}{{\int }_0^b[y{]}_0^{h}dx}$

$\overline{x}=\frac{{\int }_0^{b}xdx}{{\int }_0^{b}dx}$

Using limits $\overline{x}=\frac{{\int }_0^{b}xdx}{{\int }_0^{b}dx}$

Solving further

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

$\overline{x}=\frac{b}{2}$

Step 3: Calculation of y -

Similarly

$\overline{y}=\frac{{\iint }_{\Omega }y\rho (x,y)dA}{{\iint }_{\Omega }\rho (x,y)dA}=\frac{{\int }_0^b{\int }_0^{h}ydydx}{{\int }_0^b{\int }_0^{h}dydx}=\frac{{\int }_0^b{\left[\frac{y^2}{2}\right]}_0^{h}dx}{{\int }_0^b[y{]}_0^{h}dx}=\frac{{\int }_0^b\left[\frac{h^2}{2}\right]dx}{{\int }_0^b\left[h\right]dx}$

$\overline{y}=\frac{h{\int }_0^{b}xdx}{2{\int }_0^{b}xdx}=\frac{h{\left[\frac{x^2}{2}\right]}_0^b}{2{\left[\frac{x^2}{2}\right]}_0^b}=\frac{h\left[\frac{b^2}{2}\right]}{2\left[\frac{b^2}{2}\right]}$

$\overline{y}=\frac{h}{2}$

Hence, centroid is $\overline{x}=\frac{b}{2},\overline{y}=\frac{h}{2}$