What is the area of the region between $y=2x^2$ and $y=12-x^2$ ?

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We need to find the lower and upper limits of integration. Find the intersections of the two curves:

$$2x^2=12-x^2$$ $$3x^2 =12$$ $$x^2=4$$ $$x=2,-2$$

The area under the curve is:

$$\int\limits_{-2}^2 (12-x^2)-(2x^2)  dx = \int\limits_{-2}^2 12-3x^2  dx$$ $$= \Big[12x-x^3\Big]_{-2}^2$$ $$= (12(2)-(2)^3)-(12(-2)-(-2)^3)$$ $$(16)-(-16)= \boxed{32}$$