# Интеграл

## Обозначение

$$\displaystyle\int\limits\_{t\_1}^{t\_2}{u\~dt}$$ — определённый интеграл от $$t\_1$$ до $$t\_2$$ функции $$u(t)$$ по $$dt$$.

## Определение

$$\displaystyle\int\limits\_{t\_1}^{t\_2}{u\~dt}$$ — площадь под графиком функции $$u(t)$$ на отрезке $$\[t\_1;t\_2]$$.

<figure><img src="/files/aQncm6oLzHS8JWX7O8AM" alt=""><figcaption></figcaption></figure>

## Свойство

Если $$\dot{v}=u$$, то $$v = \displaystyle\int\limits\_{t\_1}^{t\_2}u\~dt$$.

{% code overflow="wrap" %}

```clike
import geometry;
import graph;
size(15cm);
defaultpen(fontsize(18pt));
pen gr = gray(0.8);
label("$t$", (16, 0), S);
label("$u$", (0, 8), W);
real y(real x) {
	return 4*cos((x-3)/2) + 2;
}
path g = graph(y, 2, 14);
real x1 = 3 + 4*pi / 3;
real x2 = 3 + 8*pi / 3;
path g1 = graph(y, 2, x1);
path g2 = graph(y, x1, x2);
path g3 = graph(y, x2, 14);
fill((2,0)--g1--cycle, gr);
fill((x1, 0)--g2--(x2, 0)--cycle, gr);
fill((x2, 0)--g3--(14, 0)--cycle, gr);
draw(g, linewidth(1));
draw((2, -3)--(2, 7),dashed);
draw((14, -3)--(14, 7),dashed);
path beg = shift((2, y(2))) * scale(1/8) * unitcircle;
draw(beg);
fill(beg, white);
path en = shift((14, y(14))) * scale(1/8) * unitcircle;
draw(en);
fill(en, white);
axes(min=(-1, -3), max=(16, 8), arrow=Arrow);
label("$O$", (0,0), SW);
label("$t_1$", (2, 0), SW);
label("$t_2$", (14, 0), SW);
label("$u=u(t)$", (8,6));
label("$S_1$", (3.5,3));
label("$S_2$", (9,-1));
label("$S_3$", (13,1));
label("$\displaystyle\int\limits_{t_1}^{t_2}u(t)~dt = S_1- S_2 + S_3$", (9.4, 3), fontsize(14pt));
shipout(bbox(Fill(white)));
```

{% endcode %}


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