8. Definite Integrals

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]

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) = 1/(𝓍² + 1) on [―1, 1]

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) = cos 𝓍 on [―π/2 , π/2]

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) = 𝓍ⁿ on [0,1] , for any positive integer n

Average distance on a parabola What is the average distance between the parabola y = 30𝓍 (20 ― 𝓍 ) and the 𝓍-axis on the interval [0, 20] ?

Average velocity The velocity in m/s of an object moving along a line over the time interval [0,6] is v (t) = t² + 3t. Find the average velocity of the object over this time interval.

Average height of a wave The surface of a water wave is described by y = 5 (1 + cos 𝓍) , for ― π ≤ 𝓍 ≤ π, where y = 0 corresponds to a trough of the wave (see figure). Find the average height of the wave above the trough on [ ―π , π] .

Mean Value Theorem for Integrals Find or approximate all points at which the given function equals its average value on the given interval.

ƒ(𝓍) = 8 ― 2𝓍 on [0, 4]

Mean Value Theorem for Integrals Find or approximate all points at which the given function equals its average value on the given interval.

ƒ(𝓍) = 1 ― |𝓍| on [―1, 1]