The area of the region A = (x,y) : 0 y x|x|+1, -1 x1 in sq.

Question

The area of the region $A = \{(x,y) : 0\le y\le x|x|+1,\ -1\le x\le1\}$ in sq. units is:

Accepted Answer

For $x<0$, $x|x|=-x^2$; for $x\ge0$, $x|x|=x^2$.

$$\text{Area} = \int_{-1}^0(1-x^2)\,dx + \int_0^1(1+x^2)\,dx = \left[x-\frac{x^3}{3}\right]_{-1}^0 + \left[x+\frac{x^3}{3}\right]_0^1$$

$$= \frac23 + \frac43 = 2$$

Solutions · 1