8. Definite Integrals

Find the average value of the function on the interval $[0,1]$.

$G\left(x\right)=\frac{2}{x^2+1}$

Let $f\left(x\right)=2x+3$. What is the average value of $f$ on the interval $[0,24]$?

Find the average value $f_{ave}$ of the function $f\left(x\right)=3x^2+4x$ on the interval $[-1,3]$.

Suppose ƒ is an even function and ∫⁸₋₈ ƒ(𝓍) d𝓍 = 18

(b) Evaluate ∫₋₈⁸ 𝓍ƒ(𝓍) d𝓍 .

What is the average value of the function $f\left(x\right)=x^2$ over the interval $[1,3]$?

Find the average value of the function on the interval $[4,9]$.

$F\left(x\right)=\sqrt{x}+6$

Let $f$ be the function defined by $f\left(x\right)=x$. What is the average value of $f$ on the interval $[4,6]$?

Let $f\left(x\right)=x^2+1$. What is the average value of $f$ on the closed interval $[0,2]$?

Find the average value $f_{ave}$ of the function $f\left(x\right)=x$ on the interval $[0,16]$.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(c) The average value of a linear function on an interval [a, b] is the function value at the midpoint of [a, b] .

Let $f(x,y)=x^2+y^2$. Find the average value of $f$ over the rectangle with vertices $(0,0)$, $(2,0)$, $(2,1)$, and $(0,1)$.

Find the average value of ƒ(𝓍) = e²ˣ on [0, ln 2] .

Let $f\left(x\right)=x^{\frac{1}{5}}$. What is the average value of $f$ on the interval $[0,32]$?

Average value of the derivative Suppose ƒ ' is a continuous function for all real numbers. Show that the average value of the derivative on an interval [a, b] is ƒ⁻' = (ƒ(b) ―ƒ(a))/ (b―a) . Interpret this result in terms of secant lines.

Find the average value $f_{ave}$ of the function $f\left(x\right)=3x^2+8x$ on the interval $[-1,3]$.