I am asked to find the region bounded by the following curves and I keep getting a different answer than the solutions packet. These are the curves... x = -2 ; x = 3 ; f(x) = -x^2 + 4 ; y = 0
The solutions packet says the answer is 13. Meanwhile my answer is 25/3. I really just want to know how the answer packet gets to 13....
You did,
$$\int_{-2}^{3} (-x^2+4) dx$$
But because the integral gives $\text{signed}$ area what we really want is,
$$\int_{-2}^{3} |-x^2+4| dx$$
Now note that $-x^2+4 >0$ means $4>x^2$ or $2>|x|$. This means the integrand without absolute value is already positive for $-2<x<2$. Otherwise $-x^2+4 \leq 0$. In that case because the stuff inside the absolute value is negative, we must put an extra negative sign to change it to a positive. So what we have is,
$$\int_{-2}^{2} (-x^2+4) dx+\int_{2}^{3} -(-x^2+4) dx$$
Sure enough this gives $13$.