Mathematics - Area Under Curves Question with Solution | TestHub

The required area is given by the definite integral of the function $y=|x-4|$ from $x=3$ to $x=5$.

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

Evaluating the first integral:

${\int }_3^4(4-x)dx={\left[4x-\frac{x^2}{2}\right]}_3^4=\left(16-\frac{16}{2}\right)-\left(12-\frac{9}{2}\right)=8-\frac{15}{2}=\frac{1}{2}$

Evaluating the second integral:

${\int }_4^5(x-4)dx={\left[\frac{x^2}{2}-4x\right]}_4^5=\left(\frac{25}{2}-20\right)-\left(\frac{16}{2}-16\right)=-\frac{15}{2}-(-8)=\frac{1}{2}$

Total Area $=\frac{1}{2}+\frac{1}{2}=1$ sq. unit.

Answer: $1$ sq. unit