# How do you find the area of the region between the graphs of y=x^2 and y=-x from x=0 to x=3?

Mar 25, 2017

Given two functions, f(x) and g(x), where f(x) > g(x) in a region [a,b] the area between the two functions is:

$A = {\int}_{a}^{b} \left(f \left(x\right) - g \left(x\right)\right) \mathrm{dx}$

#### Explanation:

In the region [0,3], f(x) = x^2 and g(x) =-x.

Therefore the area of the between the two function in this region is:

$A = {\int}_{0}^{3} \left({x}^{2} + x\right) \mathrm{dx}$

$A = {x}^{3} / 3 + {x}^{2} / 2 {|}_{0}^{3}$

$A = \left({3}^{3} / 3 + {3}^{2} / 2\right) - \left({0}^{3} / 3 + {0}^{2} / 2\right)$

$A = \frac{27}{2}$