Question

To three decimal places, find the average value of f(x) = 2x2 + 3 on the interval [0, 2].

Answer 1

The formula for the average value of a function is $\frac{1}{b-a} \int\limits^b_a {f(x)} \, dx$ where b is the upper bound and a is the lower.  For us, this formula will be filled in accordingly.  $\frac{1}{2} \int\limits^2_0 {(2x^2+3)} \, dx$.  We will integrate that now: $\frac{1}{2}[ \frac{2x^3}{3}+3x]$ from 0 to 2.  Filling in our upper and lower bounds we have $\frac{1}{2}[( \frac{2(2^3)}{3}+3(2))-0]$ which simplifies to $\frac{1}{2}( \frac{16}{3}+6)$ and $\frac{1}{2}( \frac{16}{3}+ \frac{18}{3})= \frac{1}{2}( \frac{34}{3})$ which is 17/3 or 5.667