Evaluate the triple integral ∫∫∫_E (xyz)dV where E is the solid:
0<=z<=5, 0<=y<=z, 0<=x<=y.
To evaluate the triple integral ∫∫∫_E (xyz)dV, where E is the solid defined by 0<=z<=5, 0<=y<=z, and 0<=x<=y, we need to rewrite the integral limits and then perform the integration.
Let's start by analyzing the integral limits for each variable:
For the z-coordinate: 0<=z<=5
For the y-coordinate: 0<=y<=z
For the x-coordinate: 0<=x<=y
Visualizing the region E in 3D, we can see that it is a pyramid-like shape, where the base lies in the xy-plane and the apex is at (0,0,5).
Since the integration limits for each variable depend on the previous ones, we will integrate in this order: x, y, and then z.
Let's start by integrating with respect to x:
∫∫∫_E (xyz)dV = ∫∫[0,z] ∫ [0,y] xy dz dy dx
Now we integrate with respect to y:
∫∫[0,z] ∫ [0,y] xy dz dy dx = ∫[0,z] ∫[0,z] xy dy dz
Finally, we integrate with respect to z:
∫[0,z] ∫[0,z] xy dy dz = ∫[0,5] ∫[0,z] xy dy dz
Applying the limits of integration:
∫[0,5] ∫[0,z] xy dy dz = ∫[0,5] [(x/2)y^2] |_0^z dz
Simplifying:
∫[0,5] [(x/2)y^2] |_0^z dz = ∫[0,5] (x/2)z^2 dz
Integrating with respect to z:
∫[0,5] (x/2)z^2 dz = (x/2) * [(1/3)z^3] |_0^5
Simplifying further:
(x/2) * [(1/3)z^3] |_0^5 = (x/2) * (1/3)(5^3 - 0^3)
Calculating:
= (x/2) * (1/3) * (125)
= (x/2) * (125/3)
= (125/6) * x
Therefore, the value of the triple integral ∫∫∫_E (xyz)dV, where E is the solid defined by 0<=z<=5, 0<=y<=z, and 0<=x<=y, is (125/6) * x.