Usually, integration using rectangles is the first step for learning integration. At its most basic, integration is finding the area between the x axis and the line of a function on a graph - if this area is not "nice" and doesn't look like a basic shape (triangle, rectangle, etc.) that we can easily calculate the area of, a good way to approximate it is by using rectangles.
Let's take an example:
This is a function that finds the area between x=0 and x=7 underneath the line of #f(x)=(3x)/2#. We already know that this is simply going to be a right-angled triangle with base 7 and height 10.5, so the area is going to be #(7*10.5)/2 = 73.5/2=36.75#.
Imagine, now, that we didn't have a formula for calculating the area of a triangle, but we did have a formula for calculating the area of a rectangle (which is base x height, as usual). Imagine that we want to "guess" the area of that triangle. So we draw ten rectangles, each with base 0.7. The height is whatever the value of x is at the left-hand side of that rectangle. So:
- The rectangle between 0 and 0.7 has height 0, because f(x) = 0 on the left-hand side.
- The rectangle between 0.7 and 1.4 has height 1.05, because f(x) = 1.05 on the left-hand side.
- The rectangle between 1.4 and 2.1 has height 2.1, because f(x) = 2.1 on the left-hand side.
- And so on.
Eventually, we'll get ten rectangles. We can easily calculate the areas of these rectangles:
- The first rectangle has area #0.7 * 0 = 0#. Remember that the base of all these rectangles is 0.7
- The second rectangle has area #0.7 * 1.05 = 0.735#.
- The third rectangle has area #0.7 * 2.1 = 1.47#.
- And so on.
The areas of the ten rectangles, in order, are:
#0, 0.735, 1.47, 2.205, 2.94, 3.675, 4.41, 5.145, 5.88, 6.615#
To get the total area of all the rectangles, we just add them together to get #33.705#. This is a pretty good approximation for our area above.
Note: you don't always have to choose the left-hand side. You can choose the right-hand side, or the midpoint, but make sure you do the same thing for the entire question - don't take one rectangle from the left-hand side and one from the right.
Also note: the smaller the rectangles, the more precise your answer!