Answer
The statement is False.
Work Step by Step
The statement is False.
Since, the average value of a function $f $ on the interval $[a,b]$ is
$$
\frac{1}{b-a}\int_{a}^{b} f(x)dx
$$
So, the average value of the function $2x^{2}+3 $ on $[1,4]$ is
$$
\begin{aligned}
\frac{1}{4-1}\int_{1}^{4} ( 2x^{2}+3 )dx &=\frac{1}{3}\int_{1}^{4} ( 2x^{2}+3 )dx.\\
& \ne \frac{1}{3}\int_{1}^{4} \pi ( 2x^{2}+3 )dx
\end{aligned}
$$
Therefore, the statement is False.
