Question

The volume of the solid \[ \left\{ (x, y, z) \in \mathbb{R}^3 \mid 1 \leq x \leq 2, 0 \leq y \leq 2/x, 0 \leq z \leq x \right\} \] is expressible as

• A. \( \int_1^2 \int_0^{2/x} \int_0^x \, dz \, dy \, dx \)
• B. \( \int_1^2 \int_0^x \int_0^{x^2} \, dy \, dz \, dx \)
• C. \( \int_1^2 \int_1^2 \int_0^x \, dz \, dy \, dx \)
• D. \( \int_1^2 \int_0^1 \int_0^{x^2} \, dy \, dz \, dx \)

Hint:

When setting up triple integrals, carefully consider the limits of integration based on the given inequalities for each variable.

Answer 1

The Correct Option is A, B, D

Step 1: Understanding the region.
The solid is defined by the constraints on \( x \), \( y \), and \( z \). The limits for \( x \), \( y \), and \( z \) are given as \( 1 \leq x \leq 2 \), \( 0 \leq y \leq 2/x \), and \( 0 \leq z \leq x \).
Step 2: Set up the triple integral.
The volume is given by the triple integral over the region: \[ V = \int_1^2 \int_0^{2/x} \int_0^x \, dz \, dy \, dx. \] This correctly represents the volume of the solid.
Step 3: Analyzing the other options.
The other options (B) and (D) can also be valid representations of the volume when reinterpreted and rearranged based on the different order of integration. Thus, the correct answers include options (A), (B), and (D).
Step 4: Conclusion.
Thus, the correct answers are (A), (B), and (D).