## Question

###### What is Integration using rectangles?

## Answers

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 toapproximateit is by using rectangles.Let's take an example:

#int_0^7(3x)/2dx# 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 sideof 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!