Changing the Order of Integration in Triple Integrals

Changing the order of integration can make a triple integral easier to evaluate without changing the solid region being measured. In this example, the solid is enclosed by a flat base, a slanted plane, two vertical planes, and a parabolic cylinder. The goal is to integrate e^x over this three-dimensional region, so the main task is to describe the boundaries in an order that leads to the simplest calculation.

One possible setup uses x as the innermost variable. In this arrangement, each line segment inside the solid runs in the x-direction. The lower limit for x comes from a vertical plane, while the upper limit comes from the parabolic cylinder. Because the parabolic boundary must be solved for x, the upper limit involves a square root.

Although this setup correctly represents the solid, it is not the most convenient. Since the integrand is e^x, integrating first with respect to x may produce expressions involving the square root boundary in later steps. This makes the remaining integrations more difficult.

To simplify the process, the order of integration is changed so that z becomes the innermost variable. With this order, each vertical segment runs straight upward from the flat base to the slanted plane. These bounds are usually easier to express because they come directly from the lower and upper surfaces of the solid.

The solid is then projected onto the xy-plane. This projection is bounded by a vertical line, a horizontal line, and a parabolic curve. In the new description, x ranges from 0 to 1, while y ranges from the parabolic curve up to the horizontal line.

This new order keeps the same three-dimensional region but describes it more efficiently. By choosing limits that match the geometry of the solid, the triple integral becomes easier to set up and evaluate.