Question

Use cylindrical coordinates. Evaluate ∫∫∫(x dV), where E is enclosed by the planes z = 0 and z = x + y + 10 and by the cylinders x² + y² = 16 and x² + y² = 25.

Answer 1

To evaluate the given triple integral, we can use cylindrical coordinates and set up an integral over the given region. By determining the bounds of integration for each variable and applying the appropriate transformation, we can calculate the result.

Step-by-step explanation:

To evaluate the given triple integral ∫∫∫(x dV) using cylindrical coordinates, we first need to determine the bounds of integration for each variable. The given region, E, is enclosed by the planes z = 0 and z = x + y + 10, and by the cylinders x² + y² = 16 and x² + y² = 25.

In cylindrical coordinates, x = rcos(θ), y = rsin(θ), and z = z. The bounds for r can be determined from the given cylinders: 4 ≤ r ≤ 5. The bounds for θ can be taken as 0 ≤ θ ≤ 2π, covering the entire circle. The bounds for z can be found by setting the equations of the planes equal to each other: z = 0 and z = rcos(θ) + rsin(θ) + 10.

Now, we can set up the integral: ∫∫∫(x dV) = ∫∫∫(rcos(θ) * r dz dr dθ) over the given bounds. We integrate first with respect to z, then r, and finally θ. Evaluating this triple integral will give us the desired result.