Final answer:
To solve the triple integral by changing to cylindrical coordinates, one must rewrite the variables and limits in terms of r, θ, and z, include the Jacobian (r), and perform the integration with respect to r, θ, and then z.
Step-by-step explanation:
The given integral ∫−2∫−√(4-y²)⁾√(4-y²) ∫∙√(x²+y²)² 2xzdzdxdy is to be evaluated by changing to cylindrical coordinates. In cylindrical coordinates, the variables x, y, z change to r, θ, z where x = rcos(θ), y = rsin(θ), z remains the same, and dx dy becomes r dr dθ. Evaluating the integral involves identifying the bounds in these new coordinates and modifying the integral accordingly. After the change of variables, the limits of integration will be based on circular symmetry and the conditions given by the original limits of x and y.
To evaluate the integral in cylindrical coordinates, the steps include:
- Identifying the new limits of integration for r, θ, and z.
- Writing the integrand in terms of r, θ, and z, including the Jacobian determinant of the coordinate transformation (r when changing dxdy to rdrdθ).
- Carrying out the integration in the new order: ∫∫∫ f(r,θ,z) r dr dθ dz.