Question

Calc II Questions

Sketch the region enclosed by the given curves and find its area.
y = 12- x^2 , y = x^2 - 6

Answer 1

72

Explanation:

Find intersection points

`12-x^2=x^2-6\\18-x^2=x^2\\18=2x^2\\9=x^2\\x=-3,3`

Set up integral and evaluate

`\displaystyle A=\int^3_(-3)\biggr((12-x^2)-(x^2-6)\biggr)\,dx\\\\A=2\int^3_0(18-2x^2)\,dx\\\\A=2\biggr(18x-(2)/(3)x^3\biggr)\biggr|^3_0\\\\A=2\biggr(18(3)-(2)/(3)(3)^3\biggr)\\\\A=2\biggr(54-(2)/(3)(27)\biggr)\\\\A=2\biggr(54-(54)/(3)\biggr)\\\\A=2(54-18)\\\\A=2(36)\\\\A=72`