Mathematics - Area Under Curves Question with Solution | TestHub

The given curves are $y=1-|x|$ and $y=|x|-1$.

To find the points of intersection, we equate the two equations:

$1-|x|=|x|-1$

$2|x|=2$

$|x|=1\Rightarrow x=\pm 1$

The area between the curves is given by the integral:

$A={\int }_{-1}^1((1-|x|)-(|x|-1))dx$

$A={\int }_{-1}^1(2-2|x|)dx$

Since the integrand $2-2|x|$ is an even function, we can write:

$A=2{\int }_0^1(2-2x)dx$

$A=2{\left[2x-x^2\right]}_0^1$

$A=2(2-1)=2$ square units.

Alternatively, the region bounded by these curves is a square with vertices at $(1,0)$, $(0,1)$, $(-1,0)$, and $(0,-1)$. The area of this square is $4\times \left(\frac{1}{2}\times 1\times 1\right)=2$ square units.

Answer: $2$