Triple Integrals in Rectangular Coordinates Triple integrals represent a three-dimensional region in the domain of three variables, representing a set of poi...
Triple integrals represent a three-dimensional region in the domain of three variables, representing a set of points whose coordinates satisfy the inequalities defined by the three variables. They provide a generalization of double integrals in two dimensions, where a single double integral represented a two-dimensional area.
Let's consider the following integral in the domain (0\le x\le 2) and (0\le y\le 3):
Interpretation: This integral represents the sum of the areas of all the rectangles with sides running from 0 to 2 in the (x) direction and from 0 to 3 in the (y) direction.
Formal Definition: A triple integral in the domain (\mathbb{R}^3) is a function of three variables (x, y, ) and (z) defined on a rectangular region (R\subset \mathbb{R}^3) by the inequalities:
\begin{split} a_1 & \le x \le b_1 \\\ a_2 & \le y \le b_2 \\\ a_3 & \le z \le b_3 \end{split}
The triple integral is then evaluated by integrating the function over the domain (R):
where (dV) is the volume element in the three-dimensional space.
Examples: