An integral is a mathematical way of adding infinitely many small contributions to find a total. What changes is the space over which those contributions are accumulated.
• Single integral: accumulates f(x) along an interval.
• Line integral: accumulates a quantity along a curve.
• Double integral: accumulates f(x,y) across a 2D region.
• Triple integral: accumulates f(x,y,z) throughout a 3D volume.
• Flux integral: measures how a vector field passes through a surface.
• Lebesgue integral: generalizes integration using a measure, making it powerful for modern analysis.
The key is simple: the domain determines what the integral measures.