Answer
$$(a)\frac{64}{3}$$
$$(b) \frac{128}{3}$$
$$(c) -\frac{64}{3}$$
$$(d) \frac{192}{3} = 64$$
Work Step by Step
(a) We are asked to find $\int_{-4}^{0} x^2 dx$ using the properties of even functions.
We are given that $\int_{0}^{4} x^2 dx = \frac{64}{3}$ and by knowing that the area to the left of the x-axis is the same as the area to the right of the x-axis, we can say that $\int_{-4}^{0} x^2 dx = \int_{0}^{4} x^2 dx$, which equals $\frac{64}{3}$.
(b) We are asked to find $\int_{-4}^{4} x^2 dx$. Because $x^2$ is an even function, we simply need to double the answer which we found in part (a). $\int_{-4}^{4} x^2 dx = 2 \int_{0}^{4} x^2 dx$, which gives us an answer of $\frac{128}{3}$.
(c) We are asked to find $\int_{0}^{4} -x^2 dx$. We simply need to pull the negative out in front, giving us an answer of $-\frac{64}{3}$.
(d) We are asked to find $\int_{-4}^{0} 3x^2 dx$. We can pull the three out in front of the integral (it is a constant), and then apply the same procedure as in part (a). $3\int_{-4}^{0} x^2 = 3\int_{0}^{4} x^2 dx$, which gives us an answer of $\frac{192}{3}$, or rather, $64$.
