Question

Use a triple integral to find the volume of the given solid. The tetrahedron enclosed by the coordinate planes and the plane 11x + y + z = 3

Answer 1

To find the volume of the given tetrahedron, use a triple integral. Set up the triple integral using the limits of integration for each variable. The volume is obtained by evaluating the triple integral.

Step-by-step explanation:

The volume of the given tetrahedron can be found using a triple integral. The tetrahedron is bounded by the coordinate planes (x=0, y=0, z=0) and the plane 11x + y + z = 3. To set up the triple integral, we need to find the limits of integration for each variable.

Let's start with the x-coordinate. Since the tetrahedron is bounded by the coordinate planes, the lower limit of integration for x is 0. To find the upper limit, we need to solve the equation of the plane for x: 11x = 3 - y - z. This gives us x = (3 - y - z) / 11.

Next, let's move on to the y-coordinate. Since the tetrahedron is bounded by the coordinate planes, the lower limit of integration for y is 0. To find the upper limit, we need to solve the equation of the plane for y: y = 3 - 11x - z. This gives us y = 3 - 11x - z. Lastly, the lower limit of integration for z is 0 and the upper limit is given by the equation of the plane: z = 3 - 11x - y.

Putting it all together, the triple integral to find the volume of the tetrahedron is:

∫0(3 - y - z)/11 ∫03 - 11x - z ∫03 - 11x - y 1 dz dy dx

Answer 2

To find the volume of the tetrahedron, set up a triple integral over the region with appropriate limits of integration.

Step-by-step explanation:

To find the volume of the tetrahedron enclosed by the coordinate planes and the plane 11x + y + z = 3, we can set up a triple integral over the region of the tetrahedron.

The limits of integration are determined by the intersection points of the plane with the coordinate planes. By setting each coordinate plane equal to zero, we find the points (3/11, 0, 0), (0, 3, 0), and (0, 0, 3) on the plane. These points will serve as the limits for our triple integral.

Based on the orientation of the tetrahedron, the volume integral becomes: V = ∫∫∫ dV where the limits of integration for x, y, and z are 0 to 3/11, 0 to 3, and 0 to 3, respectively.