Question

Find the volume of the given solid. Enclosed by the paraboloid z = x2 + y2 + 1 and the planes x = 0, y = 0, z = 0, and x + y = 7.

Answer 1

To find the volume of the given solid enclosed by the paraboloid and planes, set up a triple integral with appropriate limits and evaluate it.

Step-by-step explanation:

To find the volume of the solid enclosed by the paraboloid z = x2 + y2 + 1 and the planes x = 0, y = 0, z = 0, and x + y = 7, we need to set up the integral with appropriate limits. We can rewrite the equation of the paraboloid as z = r2 + 1, where r = sqrt(x2 + y2). The limits for x and y are given by the equation of the plane x + y = 7, so we have x = 7 - y. The volume can be calculated as:

V = ∫∫∫ (r2 + 1) rdrdθdz

where the limits of integration are:

0 ≤ z ≤ 7 - r

0 ≤ r ≤ √(7 - y)

0 ≤ θ ≤ 2π

Answer 2

The volume of the solid enclosed by the paraboloid z = x^2 + y^2 + 1 and the planes x = 0, y = 0, z = 0, and x + y = 7 is found using a double integral set up over the triangular region in the xy-plane.

Step-by-step explanation:

The volume of the solid enclosed by the paraboloid z = x^2 + y^2 + 1 and the planes x = 0, y = 0, z = 0, and x + y = 7 can be found using multiple integration. Since we are given specific boundaries, we can set up a double integral to calculate the volume. The paraboloid opens upward, and the planes form a triangular region in the xy-plane with vertices at (0,0), (7,0), and (0,7).

We first integrate with respect to y from 0 to 7-x and then with respect to x from 0 to 7. The limits are due to the triangular region formed by the plane x + y = 7 in the xy-plane. Our integral becomes:

∫07 ∫07-x (z) dy dx, where z = x^2 + y^2 + 1. The integral expression becomes ∫07 ∫07-x (x^2 + y^2 + 1) dy dx. Carrying out this integration will yield the volume of the solid.

So, the volume of the solid is 200/3 cubic units.