Name: setting triple integral bounds for easier evaluation
Uploaded: 2026-06-08T09:38:37.891Z
Duration: 1 min 26 s
Channel: JoVE

Consider a three-dimensional region bounded by a flat base, a slanted plane, two vertical planes, and a parabolic cylinder. The goal is to evaluate the triple integral of e to the power of x over this region.First, the integral is set up with x as the innermost variable. Here, x varies from the vertical plane to the parabolic cylinder, so the upper limit is written using a square root.This setup is less convenient because integrating first with respect to x introduces a square root, which makes the remaining steps harder.To simplify the setup, the order of integration is changed so that z becomes the innermost variable. The new inner limits move straight up from the flat base to the slanted plane.Next, the x- and y-limits are determined from the base of the solid in the xy-plane. This base is bounded by x equals zero, y equals one, and the parabolic curve. Here, x runs from zero to one, while y runs from x squared to one.With this new order, the integration becomes simpler. The region stays the same, but its boundaries are easier to describe after changing the order of integration.

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 ex 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 ex, 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.

Changing the Order of Integration in Triple Integrals

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Education

JoVE Core

Calculus

Unit

Multiple Integrals and Applications

KEY TERMS AND CONCEPTS

setting triple integral bounds for easier evaluation

Triple integrals can be easier to evaluate when the bounds are chosen to match the shape of the solid. In this example, the region is enclosed by a flat base, a slanted plane, two vertical planes, and a parabolic cylinder. The task is to integrate e x over this three-dimensional region, so the goal is to describe the limits in the simplest useful order.

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

That setup is correct, but it is not the easiest to work with. Since the integrand is e x, integrating first with respect to x can leave square-root terms that make later steps harder. To simplify the calculation, the order of integration is changed so that z becomes the innermost variable.

With z inside, each vertical segment runs straight upward from the flat base to the slanted plane. These limits are often easier to write because they come directly from the lower and upper surfaces of the solid. The region is then projected onto the xy-plane, where the boundary is set by a vertical line, a horizontal line, and a parabolic curve. In the new description, x ranges from 0 to 1, and y ranges from the parabolic curve up to the horizontal line. This keeps the same solid but gives a more efficient way to set up and evaluate the triple integral.

Setting Triple Integral Bounds for Easier Evaluation

setting-triple-integral-bounds-for-easier-evaluation

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changing the order of integration in double integrals

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Triple integrals can be easier to evaluate when the bounds are chosen to match the shape of the solid. In this example, the region is enclosed by a flat base, a slanted plane, two vertical planes, and a parabolic cylinder. The task is to integrate e x over this three-dimensional region, so the goal is to describe the limits in the simplest useful order. One setup uses x as the innermost variable. In that case, each line segment inside the solid runs in the x-direction. The lower limit for x...

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triple-integrals-in-cylindrical-coordinates

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key terms and concepts

Learn how changing the order of integration can simplify a triple integral by using the solid’s planes, parabolic cylinder, and xy-plane projection.

Setting Triple Integral Bounds for Easier Evaluation | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Setting Triple Integral Bounds for Easier Evaluation

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