1.1 Approximating Areas (p. 5 – 20)
2.1 Areas Between Two Curves (p. 122 – 128)

P. 21: 1 – 7 odd, 12, 15, 16, 17 (optional)
P. 131: 1 – 7 all, 11, 15 – 21 all, 23
P. 271: 63

Webwork: Riemann Sums (optional), Area Between Curves due 5/10

Notes from lecture and office hours:


1.1. Riemann Sums

The Webwork set for Riemann Sums is officially optional, but the pictures in this section will help set us up to understand how integrals can be used to compute volumes, which is the last big topic in the course.

Back at the beginning of the course, when we were setting up definite integrals to compute signed areas, there was one detail in Section 1.2 that was a little difficult to discuss because we hadn’t yet seen Section 1.1. Look back at the first definition in Section 1.2. Looks terrible, doesn’t it?! Now we’ll make sense of it using pictures.

The point of Section 1.1 is that we know how to compute areas of basic shapes, like rectangles, but not of irregular shapes shapes, like the area under a curve y=f(x). So we’ll chop up our irregular shape into a bunch of skinny pieces and approximate the area of each skinny piece by the area of a skinny rectangle. The area of a rectangle is the product of its base and its height.

Check out an example in Desmos here.

The base of each rectangle will be represented by \Delta x. To figure out the height of the i-th rectangle, we’ll let x_i^* represent any x-value in the i-th interval. Then the height of that rectangle is represented by f(x_i^*), so the area is represented by f(x_i^*) \Delta x.

If we have n rectangles, then the sum of their areas is \sum_{i=1}^n f(x_i^*) \Delta x. This is what’s known as a Riemann sum.

But this just gives an approximation of the area we’re interested in. How can we get a better approximation? Chop the shape up into more and more skinnier rectangles! Try messing with the slider for n in the above Desmos example. As n gets bigger, the approximation gets better.

What would be “the best” approximation? Well, if we could let the number of rectangles go to infinity (so that each rectangle becomes infinitely skinny) then we’d get the *actual* area under the curve.

This is why the first definition in Section 1.2 says that

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

The notation is suggestive: \sum is the capital Greek letter sigma or S, which stands for “sum;” the integral symbol \int is like an elongated “S” which stands for…guess what: “sum.” That’s not all. The symbol \Delta is the capital Greek letter delta or D, which stands for “difference;” the symbol d is just our regular lower-case Roman letter d, which stands for…guess what: “difference.” So when n goes to \infty, the \sum becomes a \int and the \Delta becomes a d.

You can visualize more examples withΒ  this applet.

2.2 Area between curves

You actually don’t *have* to know how to compute Riemann sums to calculate the area between two curves. You just have to remember that a definite integral is a signed area:

\int_a^b f(x) dx represents the (signed) area of the region between x=a and x=b and between the graph y=f(x) and the x-axis.

What if we have two functions f(x) and g(x) and we want to know the area between them?

Here is a motivating example, that illustrates pretty much everything you need to know for this section. There are three regions: a red one, a blue one, and a purple one. You want to know the area of the purple one. How can you find the area of the purple region in terms of the areas of the red region and the blue region? How can you find the areas of the red region and the blue region?

As usual, the mathispower4u account has lots of great videos showing examples. Here is just one.

The upshot is that to find the area between two curves you have to set up a definite integral…and then you have to evaluate it.