Integraalialueen

Integraalialueen refers to the region over which an integral is evaluated. In the context of multiple integrals, this is typically a two-dimensional area in the xy-plane for a double integral, or a three-dimensional volume in xyz-space for a triple integral. The shape of the integraalialueen is crucial for determining the limits of integration and the order in which the integrations are performed.

For a double integral of a function f(x, y) over a region R, the integraalialueen is the area R. This region can be described in various ways, such as being bounded by specific curves or lines. For instance, R might be a rectangle defined by $a \le x \le b$ and $c \le y \le d$, or it could be a more complex shape like the area between two functions $g_1(x) \le y \le g_2(x)$ over an interval $a \le x \le b$. The choice of how to describe the integraalialueen often depends on which description leads to simpler integration.

Similarly, for a triple integral of a function f(x, y, z) over a region V, the integraalialueen is the volume V. This region can be defined by inequalities involving x, y, and z. For example, V could be a rectangular box, or it might be a sphere, a cylinder, or a region bounded by various surfaces. Understanding the geometry of the integraalialueen is essential for setting up the correct iterated integral. The evaluation of the integral proceeds by integrating with respect to one variable at a time, with the limits of integration determined by the boundaries of the integraalialueen.