Triple Integrals in Rectangular Coordinates

A triple integral gives the total accumulation of a quantity over a three-dimensional region using a function of three variables.

Consider a solid metal block shaped like a rectangular box. At each point inside the block, the function gives the material’s density. The total mass is the accumulation of that density throughout the entire solid.

To calculate this mass, the block is divided into many sub-boxes, each with a small volume. A sample point in each sub-box is used to estimate the density in that region. Multiplying this density by the small volume gives an approximate mass for that sub-box.

Adding the masses of all the sub-boxes gives an approximate total mass called a triple Riemann sum. As the block is divided into smaller and smaller sub-boxes, the triple Riemann sum approaches the triple integral.

A triple integral can also be evaluated as an iterated integral, where accumulation happens one variable at a time. Fubini’s Theorem states that for a continuous function over a bounded region, with well-defined limits, the order of integration can be changed, and each order gives the same value.