Print preview
Handout Friday, 11/21
Today we will start section 4.4 to discover the Fundamental Theorem of Calculus. But first, from section 4.3:
One more cool thing we can do with integrals: find the average value of a function.
-
Think about what the average value of a set of numbers is. But what if we wanted to find the average value of a function. That has an infinite number of points.
-
So really what we want to compute is\begin{equation*} \lim_{n \to \infty} \frac{f(x_1) + f(x_2)+ f(x_3) + \cdots f(x_n)}{n}\text{.} \end{equation*}
-
Now compare this limit to the limit of a right Riemann sum. They are sort of similar, since \(\Delta x = \frac{b-a}{n}\text{.}\)
-
In fact, we get\begin{equation*} f_{\text{AVG}[a,b]}= \frac{1}{b-a}\cdot \int_a^b f(x)\, dx\text{.} \end{equation*}
-
Think back to what the integral tells us about change in position when integrating velocity. The other way we can think about finding change in position is by multiplying the time spent going that velocity by the velocity. But what if velocity isn’t constant? Well, multiply time by average velocity.
Baked into the formula for average value of a function is the (hopefully familiar, at this point) idea that \(\int_a^b v(t)\, dt\) gives the total change in position \(s(t)\) between \(t = a\) and \(t = b\text{,}\) where \(v(t)\) is the velocity corresponding to position function \(s(t)\text{.}\)
-
In other words:\begin{equation*} \int_a^b v(t)\,dt = s(b) - s(a)\text{.} \end{equation*}But think about the relationship between \(s\) and \(v\text{.}\) Velocity is the derivative of position. So another way we could write the equation above is\begin{equation*} \int_a^b s'(t)\, dt = s(b) - s(a)\text{.} \end{equation*}
-
In fact, for any continuous function, we can think of that function as a velocity function for some position function it is the derivative of. Big idea: if we can get our hands on that position function, we can use \(s(b) - s(a)\) to compute the integral (instead of using a limit of Riemann sums).
-
Notation: given a (continuous) function \(f\text{,}\) let \(F\) be antiderivative of \(f\) (that is, \(F'(x) = f(x)\)).
-
The Fundamental Theorem of Calculus (FTC) is that\begin{equation*} \int_a^b f(x) \, dx = F(b) - F(a)\text{.} \end{equation*}
-
Let’s use the FTC to evaluate some definite integrals. Do some examples involving constant functions, linear functions, polynomials, basic trig and exponential functions.
