Question

Calc II Question

Sketch the region enclosed by the given curves and find its area.
Y = lxl , y = x^2 - 2

Answer 1

`\displaystyle A=(20)/(3)`

Explanation:

`\displaystyle A=\int^2_(-2)(|x|-(x^2-2))\,dx\\\\A=2\int^2_0(x-(x^2-2))\,dx\\\\A=2\int^2_0(-x^2+x+2)\,dx\\\\A=2\biggr(-(x^3)/(3)+(x^2)/(2)+2x\biggr)\biggr|^2_0\\\\A=2\biggr(-(2^3)/(3)+(2^2)/(2)+2(2)\biggr)\\\\A=2\biggr(-(8)/(3)+2+4\biggr)\\\\A=2\biggr(-(8)/(3)+6\biggr)\\\\A=2\biggr((10)/(3)\biggr)\\\\A=(20)/(3)`

Bounds depend on whether you use -x or +x instead of |x|, but you double regardless. See the attached graph for a visual.