What is Integration using rectangles?

Question

Isaiah Allen · Accepted Answer

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(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 side of that rectangle. So:

Eventually, we'll get ten rectangles. We can easily calculate the areas of these rectangles:

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!

Samantha Adkison · Answer

undefined Integration using rectangles, also known as Riemann sums, is a method of approximating the area under a curve by dividing it into rectangles and summing up the areas of these rectangles. This method is foundational to the concept of definite integration.

There are three common types of Riemann sums:

1. Left Riemann Sum: The height of each rectangle is determined by the function value at the left endpoint of each subinterval.
2. Right Riemann Sum: The height of each rectangle is determined by the function value at the right endpoint of each subinterval.
3. Midpoint Riemann Sum: The height of each rectangle is determined by the function value at the midpoint of each subinterval.

As the number of rectangles increases, the approximation becomes more accurate, approaching the actual area under the curve. This process is formalized in the definition of a definite integral as the limit of Riemann sums as the width of the rectangles approaches zero.