# Integration - Area under a curve

I was working on a computer programming project that involves 2D drawing on windows OS. I was displaying curves using simple mathematical formulas, and was thinking of filling the part under a curve.

That requires to draw straight lines from x to y. Then I was thinking, summing up those lines make up the area under the curve, which is clearly what an integral in mathematics is, but I was not able to explain one thing.

The length of those lines is the corresponding y's, so basically I would be summing: y1 + y2 + y3 + ...

So, if I have a graph for y = x, the sum is:

1 + 2 + 3 + . . .

The formula for this sum is clearly ((x * x) + x) / 2, and not (x * x) / 2. I don't get it, because the laws of integration tell us that the integral of x^n is (x ^ (n+1)) / (n+1). How is that so?

• The formula for the integral does not bear resemblance necessarily to the formula for the summation. Apr 28, 2018 at 7:03
• Are you talking about Simpson's rule? Apr 28, 2018 at 7:39

What you're doing is the adding heights of the vertical blue lines on the left, which is clearly not the area under the blue curve!

Imagine you divide the interval $[0, x]$ into $N$ pieces each of size

$$\Delta = \frac{x}{N}$$

The $x$ coordinates of each rectangle on the right figure can be labeled with the number $x_i = \Delta i$. The area of each individual rectangle is

$$A_i = (x_{i + 1} - x_i) y_i = \Delta x_i = \Delta^2 i$$

So that the total area is

$$A_N = \sum_{i=1}^N A_i = \sum_{i=1}^N \Delta x_i = \frac{x^2}{N^2} \sum_{i=1}^N i = \frac{x^2}{N^2}\frac{N(N+1)}{2} = \frac{x^2}{2} \left(1 + \color{blue}{\frac{1}{N}}\right) \tag{1}$$

Look at the term $\color{blue}{1/N}$, the larger the number $N$ the smaller $1/N$. In the limit when $N\to\infty$ this number is zero, and the area converges to

$$\lim_{N\to\infty}A_N = A = \frac{x^2}{2} \tag{2}$$

• I think you are viewing the image in a different manner. The area by my definition would be the number of pixels. Just a number. It does not have to be a square. The formula x^2/2 does not calculate all the pixels under a curve, and thus does not amount to the area. Apr 28, 2018 at 9:47