Question

Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 8

Answer 1

To find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 8, we can use triple integration over the limits of the tetrahedron formed by the planes.

Step-by-step explanation:

To find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 8, we can use the concept of triple integration. Since the given planes form a tetrahedron, we'll need to integrate over the region of the tetrahedron to find its volume.

First, we need to set up the limits of integration for each variable:

• x: 0 to 8 - y - z
• y: 0 to 8 - x - z
• z: 0 to 8 - x - y

Next, we'll integrate 1 with respect to x, y, and z over the respective limits of integration:

∭1 dx dy dz = ∫08 ∫08 - x - z ∫08 - x - y 1 dz dy dx

Simplifying this triple integral will give us the volume of the solid bounded by the given planes.

Answer 2

***** sketch power in math *****

If we sketch the shape of the solid, it is a corner of a cube of side length 8.
Hence the volume is one-sixth of a cube,
Volume = 8^3/6=256/3=85.333...

Or it is a pyramid with triangular base with two faces perpendicular to the base,
area of base = 8^2/2=32, height=8, so
volume = Ab*h/3=32*8/3=256/3=85.333... as before