Q. 15 - Calculus

Question

The centroid of the region between the graphs of $f\left(x\right)=\sqrt{x}$and $g\left(x\right)=4-x$ on $\left[0, 4\right]$

Step-by-Step Solution

Answer

The centroid of the region between the two graphs $f\left(x\right)=\sqrt{x}$ and $g\left(x\right)=4-x$ in the intervals $\left[0, 4\right]$ is $\left[4, \frac{172}{25}\right]$

Step 1. Given information

Given is $f\left(x\right)=\sqrt{x}$ and $g\left(x\right)=4-x$ in the interval $\left[0, 4\right]$ 

Step 2. Calculate the centroid

$$\begin{gathered} \left(\overline{x}, \overline{y}\right)=\frac{{\int }_a^{b}x\left|f\left(x\right)-g\left(x\right)\right|dx}{{\int }_a^b\left|f\left(x\right)-g\left(x\right)\right|dx}, \frac{{\displaystyle \frac{1}{2}}{\int }_a^b\left|f\left(x\right)-g{\left(x\right)}^2\right|dx}{{\int }_a^b\left|f\left(x\right)-g\left(x\right)\right|dx} \\ {\int }_0^4\left[x^2-\left(4-x\right)\right]dx={\int }_0^4\left(x^2+x-4\right)dx \\ ={\left[\frac{x^3}{3}+\frac{x^2}{2}-4x\right]}_0^4 \\ =\frac{40}{3} \\ {\int }_0^{4}x\left[x^2-\left(4-x\right)\right]dx={\int }_0^4\left(x^3+x^2-4x\right)dx \\ ={\left[\frac{x^4}{4}+\frac{x^3}{3}-2x^2\right]}_0^4 \\ =\frac{160}{3} \\ \frac{1}{2}{\int }_0^4{\left[x^2-\left(4-x\right)\right]}^{2}dx=\frac{1}{2}{\int }_0^4\left[x^4+x^2+16+2x^3-8x^2-8x\right]dx \\ =\frac{1}{2}{\int }_0^4\left(x^4+2x^3-7x^2-8x+16\right)dx \\ =\frac{1}{2}{\left[\frac{x^5}{5}+\frac{x^4}{4}-\frac{7x^3}{3}-4x^2+16x\right]}_0^4 \\ =\frac{1376}{15} \\ {\int }_0^4\left[x^2-\left(4-x\right)\right]dx=\frac{40}{3}, {\int }_0^{4}x\left[x^2-\left(4-x\right)\right]dx=\frac{160}{3} \\ \frac{1}{2}{\int }_0^4\left[x^2-\left(4-x\right)\right]dx=\frac{1376}{15}, \\ in equation as, \\ \left(\overline{x}, \overline{y}\right)=\frac{{\displaystyle \frac{160}{3}}}{{\displaystyle \frac{40}{3}}}, \frac{{\displaystyle \frac{1376}{15}}}{{\displaystyle \frac{40}{3}}} \\ =\left[4, \frac{172}{25}\right] \end{gathered}$$

Step 3. The solution

The centroid of the region between the two graph $f\left(x\right)=\sqrt{x}$ and $g\left(x\right)=4-x$in the interval $\left[0, 4\right]$ is $\left[4, \frac{172}{25}\right]$