What is the volume of the rectangular box shown on the board?

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. Identify the side lengths of the box: 3, 4, and 2.
2. Multiply the side lengths: $3 \times 4 \times 2 = 24$.
3. Alternatively, evaluate the triple integral: $\int_{0}^{3}\int_{0}^{4}\int_{0}^{2} 1 \, dz \, dy \, dx = 24$.
4. Conclude that the volume is 24 cubic units.

Example

The speaker computes the volume geometrically as 24 cubic units. The board shows the final written value 24.

Common misconceptions

• Forgetting to multiply all three side lengths.
• Confusing the volume with the surface area.

Watch the explanation

Explore next

• What is the final volume of the rectangular box computed by the triple integral?
• Why is the integrand implicitly 1 when computing the volume of a rectangular box with a triple integral?