Final answer:
The student's question is about evaluating a triple integral in multivariable calculus. The integral evaluates to 1/2 after performing the necessary steps of nested integration, starting from the innermost integral and working outward.
Step-by-step explanation:
The student is asking to evaluate the triple integral ∫₀¹ ∫₀⁴ ∫₀²⁴ (x² + y²)dz dy dx. This is a problem in multivariable calculus involving the computation of a volume integral over a given region. The limits of each integral depend on the previous variables, which is indicative of a nested or iterated integral.
To solve this, we integrate step-by-step according to the order of dz, dy, and then dx:
- First integral with respect to z (∫ dz): ∫₀²⁴ (x² + y²)dz = z(x² + y²)|₀²⁴ = 2x(x² + y²).
- Second integral with respect to y (∫ dy): ∫₀⁴ 2x(x² + y²)dy = 2x∫₀⁴ (x² + y²)dy = 2x(x²y + y³/3)|₀⁴ = 2x^4.
- Third integral with respect to x (∫ dx): ∫₀¹ 2x⁴dx = ∫₀¹ 2x⁴dx = x⁵/2|₀¹ = 1/2.
Therefore, the value of the triple integral is 1/2.