The value of the surface integral is 3π/5.
Here's how to evaluate the surface integral:
1. Parameterization:
Since the surface S is defined by two equations, we need a parameterization to express x, y, and z in terms of two parameters.
We can use cylindrical coordinates:
θ: angle between the positive x-axis and the projection of the position vector onto the xy-plane (ranging from 0 to 2π)
r: radial distance from the origin (ranging from 0 to 3, as the cylinder has radius 3)
Therefore, the parameterization is:
x = r cos(θ)
y = r² (valid for the paraboloid)
z = r sin(θ)
2. Surface element:
The surface element in cylindrical coordinates is:
dS = r |∂z/∂θ ∂x/∂r - ∂x/∂θ ∂z/∂r| dr dθ
In our case, the partial derivatives are:
∂z/∂θ = r cos(θ)
∂x/∂r = cos(θ)
∂x/∂θ = -r sin(θ)
∂z/∂r = sin(θ)
Therefore, the surface element becomes:
dS = r |r cos²θ + r sin²θ| dr dθ = r² dr dθ
3. Integral setup:
We want to integrate over the area S defined by the cylinder and paraboloid:
∫∫S y dS = ∫₀²π ∫₀³ r² dr dθ
4. Substitution for y:
Substitute y = r² into the integral:
∫₀²π ∫₀³ r⁴ dr dθ
5. Evaluation:
Integrate with respect to r:
∫₀²π ∫₀³ r⁵/5 dr dθ = ∫₀²π 3⁶/30 dθ = 3²/10 ∫₀²π dθ
Integrate with respect to θ:
3²/10 * 2π = 3π/5
Therefore, the value of the surface integral is 3π/5.