Question

Evaluate the triple integral F(X)_{E} z d V)where ( E ) is the solid bounded by the cylinder ( y^{2}+z^{2}=324 and the planes x=0, y=3 x and z=0 in the first octant.

Answer 1

To evaluate the given triple integral, we need to find the limits of integration for x, y, and z. The solid E is bounded by a cylinder and three planes. We then integrate the function with respect to each variable within the limits to find the value of the triple integral.

Step-by-step explanation:

To evaluate the triple integral F(X)_{E} z d V, we need to find the limits of integration for each variable within the given solid E. The solid E is bounded by the cylinder (y^2 + z^2 = 324) and the planes x = 0, y = 3x, and z = 0. In the first octant, the limits of integration are:

1. 0 ≤ x ≤ 9 (from the cylinder)
2. 0 ≤ y ≤ 3x (from the plane y = 3x)
3. 0 ≤ z ≤ √(324 - y^2)} (from the cylinder)

Once we have the limits of integration, we can evaluate the triple integral using the specified function F(X) = z. We integrate z with respect to x, y, and z within their respective limits to find the value of the triple integral.