Question 1

Why would you change the order of integration in a triple integral?

Accepted Answer

Changing the order of integration simplifies the setup and evaluation without altering the solid region. In this example, integrating first with respect to x introduces a square root boundary from the parabolic cylinder, making subsequent steps harder. By reordering so z becomes the innermost variable, the bounds become simpler: z runs directly from the flat base to the slanted plane, eliminating the square root and streamlining the calculation.

Question 2

How do you determine the new integration limits when changing the order?

Accepted Answer

First, identify the geometric boundaries of the solid. When z becomes the innermost variable, it runs vertically from the lower surface to the upper surface. Then project the solid onto the xy-plane to find the outer limits. In this case, x ranges from 0 to 1, and y ranges from the parabolic curve to the horizontal line, creating a clear two-dimensional base for the remaining integrations.

Question 3

What surfaces bound the three-dimensional region in this example?

Accepted Answer

The solid is enclosed by four surfaces: a flat base at the bottom, a slanted plane at the top, two vertical planes on the sides, and a parabolic cylinder. These boundaries define the region where the triple integral of e to the power of x is evaluated. Understanding these surfaces is essential for setting up the correct integration limits in any order.

Question 4

What is the advantage of making z the innermost variable instead of x?

Accepted Answer

Making z the innermost variable produces simpler bounds because vertical segments run directly from the flat base to the slanted plane without involving square roots. Since the integrand is e to the power of x, integrating first with respect to z avoids introducing complicated expressions with square root boundaries in later steps, making the overall calculation more manageable and efficient.

Question 5

How does projecting the solid onto the xy-plane help with integration?

Accepted Answer

Projecting the solid onto the xy-plane reveals the base region bounded by a vertical line, a horizontal line, and a parabolic curve. This projection determines the outer integration limits for x and y. By visualizing this two-dimensional base, you can accurately describe how x and y vary across the region, which is essential for setting up the correct order of integration.

Question 6

Why does the original setup with x as the innermost variable create difficulties?

Accepted Answer

When x is the innermost variable, its upper limit comes from the parabolic cylinder and must be expressed as a square root. This square root boundary persists through the remaining integrations, complicating the expressions in later steps. Since the integrand is e to the power of x, this complexity makes the overall evaluation more difficult than necessary, motivating the need to change the order of integration.

Question 7

Does changing the order of integration alter the solid region being measured?

Accepted Answer

No, changing the order of integration does not alter the solid region. The same three-dimensional region bounded by the flat base, slanted plane, vertical planes, and parabolic cylinder is evaluated regardless of integration order. Only the way the boundaries are described and the sequence of calculations changes, making the process more efficient while measuring the same volume.

三重积分中积分顺序的变换