Question

Find the average value of the function y = 6 - x2 over the interval [-1, 4]

Answer 1

Answer:

The average value of the function $f(x) = 6 - x^{2}$ over the interval $[-1,4]$ is $\frac{5}{3}$.

Step-by-step explanation:

The average value of a function over an interval is represented by this integral:

$\bar y = \frac{1}{b-a}\cdot \int\limits^{b}_{a} {f(x)} \, dx$

Where:

$a$, $b$ - Lower and upper bounds of the interval, dimensionless.

$f(x)$ - Function, dimensionless.

If $a = -1$, $b = 4$ and $f(x) = 6 - x^{2}$, the average value of the function is:

$\bar y = \frac{1}{4-(-1)}\int\limits^{4}_{-1} {6-x^{2}} \, dx$

$\bar y = \frac{6}{5}\int\limits^{4}_{-1} \, dx - \frac{1}{5}\int\limits^{4}_{-1} {x^{2}} \, dx$

$\bar y = \frac{6}{5}\cdot x |_{-1}^{4} - \frac{1}{15}\cdot x^{3}|_{-1}^{4}$

$\bar y = \frac{6}{5}\cdot [4-(-1)]- \frac{1}{15}\cdot [4^{3}-(-1)^{3}]$

$\bar y = \frac{5}{3}$

The average value of the function $f(x) = 6 - x^{2}$ over the interval $[-1,4]$ is $\frac{5}{3}$.