Calculus I Lesson 20: Riemann Sums and Definite Integrals

  1. Presentations
  2. Homework Questions?
  3. Areas under curves
    1. Sigma Notation
    2. Left and Right endpoint approximations
    3. Riemann Sums
  4. Upcoming

Presentations

Homework Questions?

Areas under curves

We are now ready to start discussing areas under curves. Let’s warm up with a couple of simple examples.

Example: Suppose we are driving on the highway on cruise control at a constant speed of 55mph. If we drive for 2 hours, we have traveled a 110 miles: $55$ miles / hour $\times$ 2 hours = 110 miles.

In other words, we have this graph:

So it’s easy to find the distance traveled from our speed if the speed is constant: it’s just the area of a rectangle. But what if our speed is not constant?

Example: A particle is initially stopped. At some time $t = 0$, the particle beings to move at a rate of $2t$ meters per second (that is, the particle speeds up over time). How far has it moved after 3 seconds?

So this is no longer a simple “distance = speed $\times$ time” question. But in this case, to find the distance, we still look at an area:

This would be asking for the area of a triangle. The base of the triangle is 3, and the height is 6, and so the area is $\frac{1}{2} \cdot 3 \cdot 6 = 9$ meters. In general, after $t$ seconds, note that the distance traveled by the particle would be $\frac{1}{2} \cdot t \cdot 2t = t^2$.

So we see that questions about distance traveled over time can be turned into questions about areas under curve. And questions about areas under curves, sometimes, can be answered using geometric formulas. But what if we don’t have a nice geometric formula for this?

Example: Determine the area under the curve $y = x^2$ from $x = 0$ to $x = 1$.

How might we find the area of this region? Since the “top” side is no longer a straight line, this is not a question of geometry anymore. We need calculus tools to answer this. Our strategy will be to get better and better approximations, and come up with a limit of these approximations. That limit should be the actual area under the curve.

So how might we approximate the area? Our method is going to be to use rectangles. First, split up the interval that we want to look at into equally spaced points and draw rectangles, as below:

This is called the left-endpoint approximation. We split the interval from 0 to 1 into the four intervals: $[0, 0.25]$, $[0.25, 0.5]$, $[0.5, 0.75]$ and $[0.75, 1]$. For each one of these intervals, we looked at the rectangle we get whose height was based on the left endpoint of that interval: that is, we looked at the y-values $f(0)$, $f(0.25)$, $f(0.5)$ and $f(0.75)$. We end up with the following areas:

Adding these up, we end up with an area of about .219.

Sigma Notation

Before we continue with the strategy of getting better and better approximations, we should review “$\Sigma$ notation”.

\[\sum_{i=0}^5 (i)\]

means the sum of the terms $0 + 1 + 2 + 3 + 4 + 5$. The $i$ is called the index variable, 0 is the lower bound, and $5$ is the upper bound. We will be using this to express long sums (or sums in terms of a variable $n$ that is not known).

Left and Right endpoint approximations

Before we found an approximation of the area under $y = x^2$ using 4 rectangles. As we use more and more rectangles, our approximation will get better and better. To approximate this area using $n$ rectangles, we divide the interval $[0, 1]$ into points $x_0 = 0, x_1, x_2, \ldots, x_n = 1$, equally spaced apart. Since the length of the interval is 1, and we want $n$ rectangles, we need the distance between two points to be $\frac{1}{n}$ in this case. Let’s call that $\Delta x$. We end up with $n$ sub-intervals: $[x_0, x_1], [x_1, x_2], \ldots, [x_{n-1}, x_n]$.

Our approximation here would be $f(x_0)\Delta x + f(x_1)\Delta x + \ldots + f(x_{n-1})\Delta x$. This can be written more succinctly as $\sum\limits_{i=0}^{n-1} f(x_i) \Delta x$. This is the left-endpoint approximation, because we used the left-endpoints of each of those intervals. We could also use the right endpoints of each interval:

The right-endpoint approximation is given by $\sum\limits_{i=1}^{n} f(x_i) \Delta x$.

Riemann Sums

It turns out that there isn’t anything special about using left endpoints or right endpoints: what matters most is that the sub-intervals get smaller and smaller. As we let $n$ get larger, so we break our interval up into smaller sub-intervals, if the function is continuous then all the points on the graph in those sub-intervals will be pretty close to each other. For example, for our function $f(x) = x^2$ over the interval $[0, 1]$, if $n = 100$, we would be looking at the intervals $[0, .01], [.01, .02], [.02, .03]$, etc. If we look at the corresponding y-values, the difference between taking left endpoints and right endpoints in most of these is quite small:

$x^2$ and $(x+.01)^2$ are going to be really close, no matter what $x$ is. So rather than worrying about the specific point in each interval that we look at, what matters most is that we let $n$ get larger (use more intervals) and get better approximations. This is the idea behind a Riemann sum.

Let $f(x)$ be a function and $[a, b]$ an interval. (Usually we assume $f(x)$ is continuous over $[a, b]$). For $n$ a large positive integer, we split $[a, b]$ into $n$ intervals: $x_0 = a$, $x_1, x_2, \ldots, x_{n-1}, x_n$, with the difference between the x coordinates, $\Delta x$ defined as $\dfrac{b - a}{n}$. Then we have $n$ intervals: $[x_0, x_1], [x_1, x_2], \ldots, [x_{n-1}, x_n]$, each of length $\Delta x$. For each interval, let $x_i^{}$ be *any point in $[x_{i-1}, x_i]$. Then the area under $f(x)$ is approximated by the Riemann sum:

\[\sum_{i=1}^n f(x_i^*) \Delta x\]

As $n \rightarrow \infty$, this converges toward the actual area under the curve. That’s the definition of the definite integral (coming up next class).

\[\int_a^b f(x) dx = \lim_{n \rightarrow \infty} \sum_{i=1}^n f(x_i^*) \Delta x\]

We will see, next class, how we might actually compute this area. This is usually done using the Fundamental Theorem of Calculus, which relates the notions of areas under curves and antiderivatives.

Upcoming

  1. DeltaMath due Thursday.
  2. Rough Draft next Monday.
  3. Exam 3: December 1
  4. Guest lecture Monday 11/24: let me know if you’re coming / interested!