What is Integration using rectangles?

Oct 2, 2014

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:

${\int}_{0}^{7} \frac{3 x}{2} \mathrm{dx}$

This is a function that finds the area between x=0 and x=7 underneath the line of $f \left(x\right) = \frac{3 x}{2}$. 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 $\frac{7 \cdot 10.5}{2} = \frac{73.5}{2} = 36.75$.

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 $0.7 \cdot 0 = 0$. Remember that the base of all these rectangles is 0.7
• The second rectangle has area $0.7 \cdot 1.05 = 0.735$.
• The third rectangle has area $0.7 \cdot 2.1 = 1.47$.
• 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.