To find the average value of the function [tex]\( f(x) = x^2 + x - 2 \)[/tex] over the interval [tex]\([0, 4]\)[/tex], we'll use the formula for the average value of a continuous function [tex]\( f(x) \)[/tex] over the interval [tex]\([a, b]\)[/tex], which is given by:
[tex]\[
\text{Average value} = \frac{1}{b - a} \int_a^b f(x) \, dx
\][/tex]
In this case:
- The function [tex]\( f(x) = x^2 + x - 2 \)[/tex]
- The interval is [tex]\([a, b] = [0, 4]\)[/tex]
First, we need to find the integral of [tex]\( f(x) \)[/tex] over the interval [tex]\([0, 4]\)[/tex]:
[tex]\[
\int_0^4 (x^2 + x - 2) \, dx
\][/tex]
Based on the given information, the integral of [tex]\( f(x) \)[/tex] from 0 to 4 is:
[tex]\[
\int_0^4 (x^2 + x - 2) \, dx = \frac{64}{3}
\][/tex]
Next, we need to divide this integral value by the length of the interval to find the average value:
[tex]\[
\text{Average value} = \frac{1}{4 - 0} \int_0^4 (x^2 + x - 2) \, dx = \frac{1}{4} \cdot \frac{64}{3}
\][/tex]
Simplifying this expression:
[tex]\[
\text{Average value} = \frac{64}{3} \cdot \frac{1}{4} = \frac{64}{12} = \frac{16}{3}
\][/tex]
Thus, the average value of [tex]\( f(x) = x^2 + x - 2 \)[/tex] on the interval [tex]\([0, 4]\)[/tex] is:
[tex]\[
\boxed{\frac{16}{3}}
\][/tex]
