What is the final volume of the rectangular box computed by the triple integral?

Conditions

• The region is a rectangular box with bounds $0 \le x \le 3$, $0 \le y \le 4$, and $0 \le z \le 2$.
• The integrand is 1.

Reasoning, step by step

1. Evaluate the innermost integral: $\int_{0}^{2} 1 \, dz = 2$.
2. Evaluate the middle integral: $\int_{0}^{4} 2 \, dy = 8$.
3. Evaluate the outermost integral: $\int_{0}^{3} 8 \, dx = 24$.
4. Conclude that the volume is 24 cubic units.

Example

The speaker states the result as 24 units cubed, which is the volume of the rectangular box. The board shows the final written value 24.

Common misconceptions

• Forgetting to multiply the results of the successive integrations.
• Confusing the order of integration and applying the limits incorrectly.

Watch the explanation

Explore next

• How does the triple integral for the volume of a rectangular box reduce to a double integral?
• Why is the integrand implicitly 1 when computing the volume of a rectangular box with a triple integral?