The area of the region bounded by the curve y= |x|, x|x-2|, the x-axis and the lines x=-2 …

Question

The area of the region bounded by the curve $y=\max \{|x|, x|x-2|\}$, the $x$-axis and the lines $x=-2$ and $x=4$ is equal to\_\_\_\_\_\_\_\_\_\_

Answer: 12

Accepted Answer

Splitting by sign changes of $|x|$ and $x|x-2|$ over $[-2,4]$:

$$\text{Area} = \frac12\times2\times2 + \int_0^1(2x-x^2)\,dx + \int_1^3 x\,dx + \int_3^4(x^2-2x)\,dx$$

$$= 2 + \left[\frac{2x^2}{2}-\frac{x^3}{3}\right]_0^1 + \left[\frac{x^2}{2}\right]_1^3 + \left[\frac{x^3}{3}-x^2\right]_3^4 = 12 \text{ sq. units}$$

Solutions · 1