Question

Write down definite integrals to express the given geometric quantities :

The centroid $\left(\overline{x},\overline{y}\right)$of the region between the graph of an integrable function $f$and the x-axis on an interval $\left[a,b\right]$

Short answer

$\left(\overline{x},\overline{y}\right)=\left(\frac{{\int }_a^{b}x.f\left(x\right)dx}{{\int }_a^{b}f\left(x\right)dx},\frac{{\displaystyle \frac{1}{2}}{\int }_a^b{\left(f\left(x\right)\right)}^{2}dx}{{\int }_a^{b}f\left(x\right)dx}\right)$

Step-by-step solution

Given Information :

Given that to assume that $f\left(x\right)$is continuous and differentiable, with a continuous derivative.

Definite integral formula centroid of a region under a curve :

The centroid $\left(\overline{x},\overline{y}\right)$of the region between the graph of $f\left(x\right)$and the x-axis on the interval $\left[a,b\right]$ is

$\left(\overline{x},\overline{y}\right)=\left(\frac{{\int }_a^{b}x.f\left(x\right)dx}{{\int }_a^{b}f\left(x\right)dx},\frac{{\displaystyle \frac{1}{2}{\int }_a^b{\left(f\left(x\right)\right)}^{2}dx}}{{\int }_a^{b}f\left(x\right)dx}\right)$