f(x,y,z) = y; W is the region bounded by the plane x+y+z=2;cylinder x^2+z^2=1. and y=0.
What I have set up isL
4 int(0 to 1) int(0 to sqrt(1-x^2)) int(0 to 1-x-y) y dydzdx.
But solving this is pretty cumbersome.
Any other ideas?
To simplify the problem, you can switch to cylindrical coordinates.
In cylindrical coordinates, the equation of the plane x + y + z = 2 can be written as ρcosθ + y + ρsinθ = 2, where ρ is the distance from the z-axis and θ is the angle between the x-axis and the line connecting the point to the z-axis.
The equation of the cylinder x^2 + z^2 = 1 can be written as ρ^2cos^2θ + ρ^2sin^2θ = 1.
Since we are integrating over W, we can express the limits of integration in terms of ρ, θ, and y as follows:
ρ: 0 to 1 (distance from the z-axis)
θ: 0 to 2π (angle)
y: 0 to 2 - ρcosθ - ρsinθ (from the equation of the plane)
The integrand is given as f(x, y, z) = y = ρsinθ.
Therefore, the integral can be written as:
∫∫∫_W y dV = ∫∫∫_W ρsinθ ρ dy dθ dρ
Now we can set up the integral using the limits of integration:
∫(0 to 2π) ∫(0 to 1) ∫(0 to 2 - ρcosθ - ρsinθ) ρsinθ y dy dθ dρ
This setup in cylindrical coordinates should simplify the problem. Evaluate this integral to find the solution to the problem.