Evaluate the triple integral _E (xy)dV where E is a solid tetrahedron with vertices (0,0,0), (4,0,0), (0,1,0), (0,0,4)
I just can't seem to find the limits, of x,y and z
To evaluate the triple integral over the tetrahedron E, we need to determine the limits of integration for each of the variables x, y, and z.
Let's start by examining the limits for x.
The base of the solid tetrahedron lies on the x-y plane, and it is a right triangle with vertices (0,0,0), (4,0,0), and (0,1,0). Therefore, the range of x values is from 0 to 4.
Moving on to y, we observe that the y direction is constant throughout the solid since it does not change along the height of the tetrahedron. So, the range of y values is from 0 to 1.
Finally, let's consider the limits for z.
The triangle formed by the three vertices (0,0,0), (4,0,0), and (0,1,0) lies on the x-y plane and does not extend along the z-axis. This means that the range of z values is from 0 to some function representing the height of the tetrahedron.
To find this height, we can examine the vertex (0,0,4). This vertex lies diagonally opposite to the base triangle, and it represents the apex of the tetrahedron. The height is then the z-coordinate of this vertex, which is 4.
Therefore, the range of z values is from 0 to 4.
Putting it all together, the limits for the triple integral over the tetrahedron E are:
∫₀⁴ ∫₀¹ ∫₀⁴ (xy) dz dy dx.
Note: The order of integration can be different depending on the convenience or applicability of the problem.