Question

Evaluate the triple integral where the solid is bounded by the cylinder x² + y² = 36 and the planes y = 3x in the first octant?

Answer 1

To evaluate the triple integral, we need to determine the limits of integration based on the given information. The solid is bounded by the cylinder x² + y² = 36 and the plane y = 3x in the first octant. By evaluating the triple integral with the given limits, we can find the solution to the problem.

Step-by-step explanation:

To evaluate the triple integral, we need to determine the limits of integration based on the given information. The solid is bounded by the cylinder x² + y² = 36 and the plane y = 3x in the first octant.

Since the cylinder is given by the equation x² + y² = 36, we can rewrite it as y = sqrt(36 - x²). The plane y = 3x can be rearranged as x = (1/3)y.

Now we have the limits of integration: for x, it ranges from 0 to (1/3)y, for y, it ranges from 0 to 6, and for z, it ranges from 0 to sqrt(36 - x²).

The triple integral can then be set up as ∫∫∫ f(x,y,z) dz dy dx, where f(x,y,z) is the desired function to integrate.

By evaluating the triple integral with the given limits, we can find the solution to the problem.