What is Migration?

Step 1: Definition of integration in calculus.

Integration is the inverse process of differentiation. It finds the antiderivative (indefinite integral) or the accumulated quantity like area (definite integral) of a function.

Indefinite integral:
$\int f(x) \, dx = F(x) + C$
where $F'(x) = f(x)$ and $C$ is the constant of integration.

Example:
$\int x^2 \, dx = \frac{x^3}{3} + C$

Step 2: Definite integral and area under curve.
The definite integral from $a$ to $b$ gives the net signed area between the curve $y = f(x)$ and the x-axis:
$\int_a^b f(x) \, dx = F(b) - F(a)$
by the Fundamental Theorem of Calculus.

Example: Area under $y = x$ from $0$ to $2$:
$\int_0^2 x \, dx = \left[ \frac{x^2}{2} \right]_0^2 = \frac{4}{2} - 0 = 2$

Step 3: Notation and uses.

• $\int f(x) \, dx$: indefinite integral.
• $\int_a^b f(x) \, dx$: definite integral.
  Uses: area, volume, displacement (if $v(t)$ is velocity), total charge (if current), etc.

Final answer:
Integration is finding $\int f(x) \, dx$, the antiderivative or accumulated area/quantity.
Integration is the inverse of differentiation: \int f(x),dx = F(x) + C where F'(x) = f(x)