Evaluate the triple integral ∫∫∫_E (xy)dV where E is the solid tetrahedon with vertices (0,0,0), (4,0,0), (0,1,0), (0,0,4)
Evaluate the triple integral ∫∫∫_E (xy)dV where E is the solid tetrahedon with vertices (0,0,0), (4,0,0), (0,1,0), (0,0,4)
To evaluate the given triple integral ∫∫∫_E (xy)dV, we need to first determine the limits of integration for each variable.
Let's start with the limits of integration for z. We know that the tetrahedron has vertices (0,0,0), (4,0,0), (0,1,0), and (0,0,4). The base of the tetrahedron lies in the xy-plane, so the z-coordinate varies from 0 to the height of the tetrahedron. The height of the tetrahedron can be found by looking at the difference in the z-coordinates of the vertex (0,0,4) and (0,0,0). Therefore, the limits for z are from 0 to 4.
Next, let's determine the limits of integration for y. As we move along the y-axis, the tetrahedron's base becomes smaller due to the linear slope between the vertices (0,0,0), (4,0,0), and (0,1,0). At any given x-value, the y-coordinate varies from 0 to the y-value on the base of the triangle at that x-value. The equation of the line connecting the points (0,0) and (4,0) is y = 0. The equation of the line connecting the points (0,0) and (0,1) is y = (1/4)x. Therefore, for any given x-value ranging from 0 to 4, the limits for y are from 0 to (1/4)x.
Lastly, let's determine the limits of integration for x. The base of the tetrahedron lies in the xy-plane and it is a right triangle with vertices (0,0), (4,0), and (0,1). Therefore, the x-coordinate varies from 0 to 4.
Now, we can set up the integral:
∫∫∫_E (xy)dV = ∫[0 to 4]∫[0 to (1/4)x]∫[0 to 4] (xy)dzdydx
Evaluating this triple integral will involve integrating the innermost integral with respect to z, then the middle integral with respect to y, and finally the outermost integral with respect to x.
The result of this triple integral will give you the value of the expression (xy) over the given solid tetrahedron, with vertices (0,0,0), (4,0,0), (0,1,0), and (0,0,4).