8. Definite Integrals

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

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 $(0,1)$.

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

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

Find the average value $f_{ave}$ of the function $f\left(x\right)=3x^2+8x$ on the interval $[-1,5]$.

Let $f\left(x\right)=x^{\frac{1}{2}}$. What is the average value of $f$ on the interval $[0,16]$?

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

Find the average value $f_{ave}$ of the function $f\left(t\right)=e^{\tan \left(t\right)}\tan \left(t\right)$ on the interval $[0,\frac{\pi }{2}]$.

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

Given a function $f\left(x\right)$, what is the average rate of change of $f\left(x\right)$ over the interval $[4,13]$?

Find the average value of the function on the interval $[2,5]$.

$F\left(x\right)=x^3-\frac{2}{\sqrt{x}}$

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

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

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

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

Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.

(a) ∫₋₄⁴ ƒ(𝓍) d𝓍

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.

ƒ(𝓍) = 𝓍³ on [―1, 1]