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 ∫∫∫_E (xy) dV, we need to determine the limits of integration for x, y, and z.
First, let's determine the limits for x.
Looking at the vertices of the tetrahedron, we can see that the x-coordinate ranges from 0 to 4. Therefore, the limits for x are 0 ≤ x ≤ 4.
Next, let's determine the limits for y.
Similarly, by examining the vertices, we can see that the y-coordinate ranges from 0 to 1. So, the limits for y are 0 ≤ y ≤ 1.
Finally, we need to determine the limits for z.
For the z-coordinate, we need to consider the equation of the plane that contains the tetrahedron. The equation of the plane can be written as z = 4 - (4/4)x - (4/1)y.
To find the limits for z, we need to consider the range of z within the given x and y limits. Let's substitute the x and y limits into the equation of the plane:
When x = 4 and y = 0:
z = 4 - (4/4)(4) - (4/1)(0) = 4 - 4 - 0 = 0
When x = 0 and y = 1:
z = 4 - (4/4)(0) - (4/1)(1) = 4 - 0 - 4 = 0
Thus, the limits for z are 0 ≤ z ≤ 4 - (4/4)x - (4/1)y.
Now that we have the limits for x, y, and z, we can set up the triple integral as follows:
∫∫∫_E (xy)dV = ∫₀¹ ∫₀⁴ ∫₀^(4 - (4/4)x - (4/1)y) (xy) dz dx dy
Evaluate the above integral using the provided limits to find the result.