Average Values

Quiz Answers: (1) 29, (2) $ \frac{1}{2}\ln\left\vert x^2 + 1\right\vert + \tan^{-1}(x)$

Exam 1: Wednesday, Feb 1, 7:00pm-7:50pm, here.
Today: §6.5 - Average Values
Today: §10.3 - Polar coords
NEXT: §10.4 -Areas in Polar coords

Why did we skip from §6.5 to §10.3? Later we'll go back and look at trig functions and complex exponentials; these ideas will fit together more than you might expect. We'll go back to §7.1 on Feb 3.

In this section we use Riemann sums to extend the familiar notion of an average, which provides yet another physical interpretation of integration.

Recall: Suppose $ y_1, \ldots, y_n$ are the amount of rain each day in La Jolla, since you moved here. The average rainful per day is

$\displaystyle y_{\avg } = \frac{y_1+\cdots + y_n}{n} = \frac{1}{n}\sum_{i=1}^n y_i.

Definition 3.3.1 (Average Value of Function)   Suppose $ f$ is a continuous function on an interval $ [a,b]$. The average value of $ f$ on $ [a,b]$ is

$\displaystyle f_{\avg } = \frac{1}{b-a} \int_{a}^b f(x) dx.

Motivation: If we sample $ f$ at $ n$ points $ x_i$, then

$\displaystyle f_{\avg } \sim \frac{1}{n}\sum_{i=1}^n f(x_i)
= \frac{(b-a)}{n(b-a)}\sum_{i=1}^n f(x_i)
= \frac{1}{(b-a)}\sum_{i=1}^n f(x_i) \Delta x,

since $ \displaystyle \Delta x = \frac{b-a}{n}$. This is a Riemann sum!

$\displaystyle \frac{1}{(b-a)} \lim_{n\to\infty} \sum_{i=1}^n f(x_i) \Delta x
= \frac{1}{(b-a)} \int_{a}^b f(x) dx.

This explains why we defined $ f_{\avg }$ as above.

Example 3.3.2   What is the average value of $ \sin(x)$ on the interval $ [0,\pi]$?
Figure: What is the average value of $ \sin(x)$?
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$\displaystyle \frac{1}{\pi - 0}\int_{0}^{\pi} \sin(x)dx$ $\displaystyle = \frac{1}{\pi - 0} \Bigl[-\cos(x)\Bigr]_{0}^{\pi}$    
  $\displaystyle = \frac{1}{\pi} \Bigl[-(-1) - (-1)\Bigr]_{0}^{\pi} = \frac{2}{\pi}$    

Observation: If you multiply both sides by $ (b-a)$ in Definition 3.3.1, you see that the average value times the length of the interval is the area, i.e., the average value gives you a rectangle with the same area as the area under your function. In particular, in Figure 3.3.1 the area between the $ x$-axis and $ \sin(x)$ is exactly the same as the area between the horizontal line of height $ 2/\pi$ and the $ x$-axis.

Example 3.3.3   What is the average value of $ \sin(2x) e^{1-\cos(2x)}$ on the interval $ [-\pi,\pi]$?
Figure 3.3.2: What is the average value?
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$\displaystyle \frac{1}{\pi - (-\pi)}\int_{-\pi}^{\pi} \sin(2x) e^{1-\cos(2x)} dx = 0 \qquad ($since the function is odd!$\displaystyle )$    

Theorem 3.3.4 (Mean Value Theorem)   Suppose $ f$ is a continuous function on $ [a,b]$. Then there is a number $ c$ in $ [a,b]$ such that $ f(c) =f_{\avg }$.

This says that $ f$ assumes its average value. It is a used very often in understanding why certain statements are true. Notice that in Examples 3.3.2 and 3.3.3 it is just the assertion that the graphs of the function and the horizontal line interesect.

Proof. Let $ F(x) = \int_{a}^x f(t) dt$. Then $ F'(x) = f(x)$. By the mean value theorem for derivatives, there is $ c\in [a,b]$ such that $ f(c) = F'(c) = (F(b) - F(a))/(b-a).$ But by the fundamental theorem of calculus,

$\displaystyle f(c) = \frac{F(b) - F(a)}{b-a} = \frac{1}{b-a}\int_{a}^{b} f(x)dx = f_{\avg }.$

$ \qedsymbol$

William Stein 2006-03-15