# Visual Integral of Cosine

## Метаданные

- **Канал:** Mathematical Visual Proofs
- **YouTube:** https://www.youtube.com/watch?v=WsC_bqRT-is

## Содержание

### [0:00](https://www.youtube.com/watch?v=WsC_bqRT-is) Segment 1 (00:00 - 00:00)

Let's consider the function f ofx= cosine of x is shown here. We can build a new function by measuring the signed area between this curve and the x-axis starting at the coordinate zero. So as we let x vary, we see that the area under the curve varies and it seems to move between the values of minus1 and 1. Let's do more and plot this new area accumulation function a of x whose value is given by the definite integral from 0 to the input x of cosine of t with respect to t. Notice that the values of this new function start positive, move negative, and come back to zero over the shown range of 0 to 2 pi. We can also extend the curve in the opposite direction, but all the areas will be the opposite sign since we are traversing the opposite direction of the definite integral. This means that the area accumulation function is an odd function with symmetry about the origin. The sign function has the same properties and looks like this accumulation function. It turns out, but requires more proof, that this area accumulation function is exactly the sign function. So that the integral of cosine is essentially the sign function.

---
*Источник: https://ekstraktznaniy.ru/video/40275*