Using Integration to Find the Area Under a Line

At first, looking at a curved line, it might be hard to think of a way to calculate the area beneath it. Try thinking of a long sleeve shirt, and then being asked to calculate the surface area of that shirt. A way we could do this is by thinking of the body of the shirt as a rectangle and then the two sleeves as smaller longer rectangles. If we break the shirt up in this way, we can simply calculate the area of each section of the shirt and then add it together. We can calculate the area underneath a line with a similar method. This method of calculating is approximate.

Have a look at this graph with the line of the function y = f(x)function f(x) shown plotted on an x-y graph

We are wanting to calculate the area underneath the line y = f(x) in between the two points a and b,

Now, we can break up the area underneath the line that into sections and calculate the area of each section, then add up all the sections to get our area,function f(x) shown plotted on an x-y graph and with the area under the curve covered in a square grid

The more times we break up the area, the more accurate our area calculation will be,function f(x) shown plotted on an x-y graph and with the area under the curve covered in a square grid - a smaller grid than the previous image

Taking an even close look at the sections that have curves, we can keep applying the break up sectioning method to make our area calculation more accurate,function f(x) shown plotted on an x-y graph and with the area under the curve covered in a square grid and with an expanded view of one of the partially filled boxes.

Then, sectioning that enlarged image into sections again,function f(x) shown plotted on an x-y graph and with the area under the curve covered in a square grid and with an expanded view of one of the partially filled boxes with it's own grid.

The more sections we have of the area we want to calculate, the more accurate the calculation will be. If each of our sections has a width of w, then the smaller the width of each section is, the more sections we will have. So as our width, w, of each section gets smaller, the more accurate our area. We can also say this like, as the width, w, gets closer to zero (i.e. gets smaller), the more accurate our area.

If we refer to the approximate area we calculate using the sectioning method as Aw,

And the accurate area we want as A,

Then as the width of each section gets smaller (i.e. as w → 0) the more closely the Area calculated using the sectioning method (Aw) gets to the actual Area we want (A)function f(x) shown plotted on an x-y graph and with the area under the curve covered in a square grid and with an expanded view of one of the partially filled boxes with it's own grid with each one dimensioned with width, w.

Theoretical Definition of the Area Under a Line

So, mathematically:

A = lim[w → 0] Aw

So the actual area, A, is equal to Aw when w is at the limit of how close w can be to zero,

[w is the width of each section we have split the area underneath the graph up into]

You can also think of this method like pixelation. When an animation, video, or image has more pixels, the represented visual is more accurate. (think of a video game from the 90s vs a video game now)

This method is referred to as Integration.