Final answer:
To evaluate the triple integral y^2 dV for the given solid tetrahedron, we need to find the limits of integration for each variable. We can then set up the triple integral and evaluate it step by step.
Step-by-step explanation:
To evaluate the triple integral y^2 dV, we need to find the limits of integration for each variable. The solid tetrahedron, with vertices (0,0,0), (2,0,0), (0,2,0), (0,0,2), can be defined with the following limits:
- For x: 0 to 2
- For y: 0 to 2-x
- For z: 0 to 2-x-y
Now, we can set up the triple integral:
∫∫∫ y^2 dV = ∫ ∫ ∫ y^2 dz dy dx
Integrating y^2, we get 1/3 * y^3. Evaluating the integral with the given limits, we have:
∫ ∫ ∫ y^2 dz dy dx = 1/3 ∫ ∫ ∫ y^3 dz dy dx = 1/3 ∫ ∫ (2-x-y)^3 dy dx = 1/3 ∫ (1/2 *(2-x-y)^4) dx = 1/3 * (1/5 *(2-x-y)^5) evaluated from x=0 to x=2