Question

Use cylindrical coordinates to evaluate the triple integral

∫∫∫E√X² + y² dV,

where E is the solid bounded by the circular paraboloid z=1-9(x²+y²) and xy-plane.

Answer 1

To evaluate the triple integral ∫∫∫E√X² + y² dV using cylindrical coordinates, convert the equation of the solid to cylindrical coordinates. Determine the limits of integration for r, theta, and z. Set up and evaluate the triple integral.

Step-by-step explanation:

To evaluate the triple integral ∫∫∫E√X² + y² dV using cylindrical coordinates, we need to convert the equation of the solid from rectangular to cylindrical coordinates.

Since the solid is bounded by the circular paraboloid z = 1 - 9(x² + y²) and the xy-plane, we can rewrite the equation in cylindrical coordinates as z = 1 - 9r².

Next, we need to determine the limits of integration for r, theta, and z. The limits for r are 0 to the radius of the circular base, which is the value of r that satisfies z = 0: 1 - 9r² = 0, r = 1/3. The limits for theta are 0 to 2π, and the limits for z are from the paraboloid to the xy-plane, which are 1 - 9r² to 0.

Now, we can set up the triple integral: ∫∫∫E√X² + y² dV = ∫02π ∫01/3 ∫1-9r²0 √r² dz dr dθ. Evaluating this triple integral will give us the final result.