Mathematics - Area Under Curves Question with Solution | TestHub

The required area is bounded by $y=|x-4|$, $x=3$, $x=5$, and the X-axis.

Area $={\int }_3^5|x-4|dx$

Since $|x-4|=-(x-4)$ for $x<4$ and $|x-4|=(x-4)$ for $x\ge 4$, we split the integral at $x=4$:

Area $={\int }_3^4(4-x)dx+{\int }_4^5(x-4)dx$

Area $={\left[4x-\frac{x^2}{2}\right]}_3^4+{\left[\frac{x^2}{2}-4x\right]}_4^5$

Area $=\left((16-8)-\left(12-\frac{9}{2}\right)\right)+\left(\left(\frac{25}{2}-20\right)-(8-16)\right)$

Area $=\left(8-\frac{15}{2}\right)+\left(-\frac{15}{2}+8\right)$

Area $=\frac{1}{2}+\frac{1}{2}=1$

Answer: $1$