The integral is a method to find the area under a curve. It is formulated as a sum of many smaller areas that approximate the area of the curve, all added together to find the total area. We let the number of areas under the curve approach infinity so the approximation to the area becomes the actual area under the curve. This is called a Reimann Sum.

Integrals were created to find the area, but it was discovered that they are related to derivatives. This discovery leads to integrals being what is called an “antiderivative.” The fundamental theorem of calculus attaches the theory of derivatives and integrals together, forming what we consider modern calculus.

The Problem: How To Find Area Under A Curve?

Area has always been of interest to society. Ancient farmers needed ways to divide up the land, which ideally would be located adjacent to a river. Since rivers wind back and forth in almost unpredictable ways, ensuring that each farmer received the same area in which to grow crops, a method for determining areas was required.

In trying to solve for the area of anything, we have to ask ourselves ‘What is the meaning of area?’

If you have a square, or rectangle, or any shape with straight edges, the area is somewhat easy to calculate. But what about the farmers with the winding river? How do you choose the boundaries of a curving river to be a straight side? You would have to start taking approximations of the river using straight lines in order. I used straight lines in my crude drawing above to try and segregate the proposed farm sections. But you may be able to spot that my plots are not all the same area. Some farmers would complain!

For a rectangle, the area is found by multiplying the length and the width. The area of a triangle is half the base multiplied by the height. The area of a polygon can be discovered by compartmentalizing it into triangles and adding the areas of the triangles.

We have methods to solve the areas of straight lines. But what about curved lines? We need a precise definition of the area. Let us start with a general curve:

general curve graph

We have no way (yet) to calculate the area found underneath this curve. So to make a crude approximation, we will draw rectangles whose height is from the x-axis to the function, and whose width is chosen so there are 5 equal width rectangles.

crude approximation of integral

You can see the first few rectangles overestimate the area under the curve, while the last few underestimate it. But if we add up all the areas of the rectangles, we arrive at a simplistic approximation to the area under the curve.

$$\text{Area} = f(x_0)\times (x_1-x_0)+f(x_1)\times (x_2-x_1)+f(x_2)\times (x_3-x_2)+f(x_3)\times (x_4-x_3) + f(x_4)\times (x_5-x_4)$$

If we know that all the \(x_i\) points are the same distance apart, we can rename that distance to \(\Delta x\) where \(\Delta x_i = x_{i+1}-x_{i}\)

Then we can re-write our sum of all rectangles in summation notation.

$$ \text{Area}=\sum_{i=0}^{4} f(x_i) \Delta x_i $$

Now imagine if we have a lot more rectangles:

Hopefully, it is obvious to you that by using many more rectangles of smaller widths we have reduced the error in how much each rectangle overestimates or underestimates the height of the curve. If we continue this trend and let the number of rectangles approach infinity, and the width of the rectangles approaches zero, then the accuracy of the area under the curve becomes perfect. We write such a notion like this:

$$\text{Area}= \underset{n \to \infty}{\lim_{\Delta x \to 0}} \sum_{i=0}^{n} f(x_i)\Delta x$$

By letting the distance between the two points make the width of the rectangle go to zero, we let the number of rectangles approach infinity by using the limit. The result is a perfect representation of the area under a curve. We call this result The Reimann Sum and we give it a special name:

The Integral.


$$\text{Area}=\text{integral} = \underset{n \to \infty}{\lim_{\Delta x \to 0}} \sum_{i=0}^{n} f(x_i)\Delta x = \int_a^b f(x) dx$$

The symbol \(\int\) stands for the integral, \(a\) and \(b\) are called the bounds of integration, the \(dx\) stands for \(\Delta x\) and represents an infinitesimal amount of \(x\), and \(f(x)\) is the curve we are looking for the area under.