Final answer:
To compute the volume of U = {(x,y,): 0 ≤ z ≤ 36 - x² - y²} using integration in cylindrical coordinates, express the equation of the volume in cylindrical coordinates, determine the limits of integration based on the region in the xy-plane, and evaluate the triple integral.
Step-by-step explanation:
To compute the volume of U = {(x,y,): 0 ≤ z ≤ 36 - x² - y²} using integration in cylindrical coordinates, we can express the equation of the volume in cylindrical coordinates as z = 36 - r². The limits of integration will be determined by the region in the xy-plane where z is defined. Since z is defined from 0 to 36 - r², the limits for r will be 0 to √(36 - z) and the limits for θ (the angle) will be 0 to 2π.
Using the cylindrical coordinate transformation, we have:
∫∫∫ U dV = ∫∫∫ r dz dr dθ, with the limits of integration being:
0 to 2π for θ
0 to √(36 - z) for r
0 to 36 for z
Now, we can integrate the volume:
∫∫∫ U dV = ∫₀²π ∫₀√(36 - z) ∫₀³₆ r dz dr dθ
After evaluating the integral, we will obtain the volume of the region U.