Average $y$ from a range of $x$ in a parabola

Question

Given a parabolic/quadratic formula such as $ax^2 + bx + c =y$, how do I get the average value of $y$ given a range of $x$ ($x_{min}$ to $x_{max}$).

Real world example: if my formula represents the trajectory arc of a thrown object under the force of gravity, where $x$ is time and $y$ is height, I'm looking for the object's average height within a timespan.

Answer 1

The average of a function $f(x)$ over interval $[x_1, x_2]$ is given by:

$$\overline{f(x)} = \frac{1}{x_2 - x_1}\int_{x_1}^{x_2}f(x)dx$$

Applying that here, and assuming that you meant $y$ as a function of time ($x$ represents time, and we'll relabel it $t$ for clarity):

$$\overline{y(t)} = \frac{1}{t_2 - t_1}\int_{t_1}^{t_2} (at^2 + bt + c)dt = \frac{\frac 13a(t_2^3 - t_1^3) + \frac 12b(t_2^2 - t_1^2) + c(t_2 - t_1)}{t_2 - t_1} = \frac 13a(t_2^2 + t_2t_1 + t_1^2) + \frac 12b(t_2 + t_1) + c$$

Answer 2

The average of a function $f(x)$ in the interval $a$ to $b$ is given as $$\frac{1}{b-a} \int_a^b f(x) dx$$