Final answer:
To evaluate the given triple integral, convert to cylindrical coordinates and use the limits of integration. Simplify the integrand and evaluate the integral to find the result.
Step-by-step explanation:
To evaluate the given triple integral ∫∫∫E y² dv, where E is the solid hemisphere x² + y² + z² < 9 and y > 0, we can use cylindrical coordinates. In cylindrical coordinates, the equation of the hemisphere becomes ρ² + z² < 9 and ρ > 0. The limits of integration are ρ: 0 to 3, φ: 0 to π/2, and z: 0 to √(9 - ρ²).
The integrand y² can be written in cylindrical coordinates as y² = (ρsinφ)² = ρ²sin²φ. Therefore, the triple integral becomes ∫∫∫E y² dv = ∫0π/2 ∫03 ∫0√(9 - ρ²) ρ²sin²φ dz dρdφ.
Simplifying and evaluating the integral gives the result ∫∫∫E y² dv = 9π/20.