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, which is a tetrahedron enclosed by the coordinate planes and the plane 8x + y + z = 4, we can use a triple integral.
First, let's define the limits of integration for each variable. Since the tetrahedron is enclosed by the coordinate planes, its limits of integration are as follows:
- For x: 0 to the intersection point of the plane 8x + y + z = 4 with the x-axis.
- For y: 0 to the intersection point of the plane 8x + y + z = 4 with the y-axis.
- For z: 0 to the intersection point of the plane 8x + y + z = 4 with the z-axis.
To find these intersection points, we set each variable equal to zero in the equation of the plane:
For x: 8x + y + z = 4, if we set y = 0 and z = 0, we get 8x = 4, so x = 0.5.
For y: 8x + y + z = 4, if we set x = 0 and z = 0, we get y = 4.
For z: 8x + y + z = 4, if we set x = 0 and y = 0, we get z = 4.
Therefore, the limits of integration are as follows:
- For x: 0 to 0.5
- For y: 0 to 4
- For z: 0 to 4
Now, we can set up the integral to calculate the volume of the tetrahedron. Since we are integrating over a volume, the integrand is equal to 1. The triple integral is given by:
∭ 1 dV
To evaluate this integral, we integrate with respect to x, y, and z in that order, using the appropriate limits of integration:
V = ∫[x=0 to 0.5] ∫[y=0 to 4] ∫[z=0 to 4] 1 dz dy dx
Solving this integral will give you the volume of the tetrahedron enclosed by the coordinate planes and the plane 8x + y + z = 4.