Use a triple integral to find the volume of the given solid.
The tetrahedron enclosed by the coordinate planes and the plane
8x + y + z = 4
To find the volume of the given solid, we can use the concept of a triple integral. A triple integral allows us to integrate a function over a three-dimensional region. In this case, we will integrate the constant function 1 over the region representing the tetrahedron.
To proceed with the calculation, we need to determine the limits of integration for each variable. Let's start by examining the region defined by the tetrahedron.
The given equation is 8x + y + z = 4. By setting each variable to 0, we can find the intersection points with the coordinate planes:
Setting x = 0, we have y + z = 4, so y = 4 - z.
Setting y = 0, we have 8x + z = 4, so x = (4 - z) / 8.
Setting z = 0, we have 8x + y = 4, so y = 4 - 8x.
By graphing these equations, we can visualize that the tetrahedron is formed by the coordinate planes (x = 0, y = 0, and z = 0) and the plane 8x + y + z = 4. The tetrahedron is bounded by the planes x = 0, y = 0, z = 0, and the intersection points (0, 4, 0), (0, 0, 4), and (1/2, 0, 3/2).
Now we can set up the triple integral to find the volume:
V = ∭R dV
where R represents the region of integration. In this case, R is the tetrahedron formed by the given planes.
The limits of integration will vary for each variable. Let's consider the limits for each variable:
For x: Since the region is bounded by x = 0 and the plane 8x + y + z = 4, the limits for x are x = 0 to x = (4 - z) / 8.
For y: Since the region is bounded by y = 0 and the plane 8x + y + z = 4, the limits for y are y = 0 to y = 4 - 8x.
For z: Since the region is bounded by z = 0 and the plane 8x + y + z = 4, the limits for z are z = 0 to z = 4 - 8x - y.
Now we can rewrite the volume integral using these limits:
V = ∭R dV = ∫[0 to (4 - y) / 8] ∫[0 to 4 - 8x] ∫[0 to 4 - 8x - y] dz dy dx
To evaluate this triple integral, you can follow these steps:
1. Integrate with respect to z, treating x and y as constants.
∫[0 to 4 - 8x - y] dz = z |[0 to 4 - 8x - y] = 4 - 8x - y
2. Integrate the result from step 1 with respect to y, while keeping x as a constant.
∫[0 to 4 - 8x] (4 - 8x - y) dy = (4 - 8x)y - (1/2)y^2 |[0 to 4 - 8x]
= (4 - 8x)(4 - 8x) - (1/2)(4 - 8x)^2
3. Integrate the result from step 2 with respect to x.
∫[0 to (4 - z) / 8] (4 - 8x)(4 - 8x) - (1/2)(4 - 8x)^2 dx
After performing this calculation, you will find the volume of the tetrahedron enclosed by the coordinate planes and the given plane.