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 graphs of two integrable functions $f\left(x\right)$ and $g\left(x\right)$ on an interval $\left[a,b\right]$.

Step-by-Step Solution

Answer

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

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 between two curves :

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

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