Multiple integrals are powerful extensions of single variable integration to functions of two, three, or more variables. While a single integral calculates the area under a curve, a double integral calculates the volume under a surface over a two-dimensional region. Triple integrals extend this concept further to calculate quantities like mass or volume in three-dimensional space.
A double integral represents the volume under a surface z equals f of x comma y over a region R in the xy-plane. The notation uses two integral signs and dA represents the differential area element. We can evaluate this as an iterated integral, integrating first with respect to one variable, then the other. For example, the integral of x squared plus y squared over a rectangular region gives us the volume under this paraboloid.
To evaluate double integrals, we use iterated integration. This means we integrate with respect to one variable at a time, treating the other as a constant. The limits of integration depend on the region. For a triangular region where y ranges from 0 to x, we first integrate with respect to y from 0 to x, then integrate the result with respect to x. The vertical line shows how y varies for a fixed x value.
Triple integrals extend integration to three dimensions, allowing us to calculate volumes of three-dimensional regions and masses of objects with variable density. The notation uses three integral signs and dV represents the differential volume element. We can evaluate triple integrals as iterated integrals, integrating with respect to one variable at a time. This cube represents a simple three-dimensional region where we might integrate a function like xyz over the entire volume.
To summarize what we have learned about multiple integrals: They are powerful extensions of single variable integration that allow us to work with functions of multiple variables. Double integrals calculate volumes under surfaces over two-dimensional regions, while triple integrals extend this to three dimensions for calculating volumes and masses. We evaluate these using iterated integration, one variable at a time. These tools have wide applications in physics, engineering, and probability theory.