Q. 7 - Calculus

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 $\left[a,b\right]$ 

Step-by-Step Solution

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 1: Given Information :

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

Step 2: 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 

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