Question

Evaluate the triple integrals:
∫∫∫xydV, where Q is the solid tetrahedron with
Q
vertices (0,0,0), (1,0,0), (0,2,0), and (0,0,3).
Please explain step by step

Answer 1

Here is an example straight from Calc book. EXAMPLE 2 Finding the Limits of Integration in the Order dy dz dx
Set up the limits of integration for evaluating the triple integral of a function F(x, y, z) over
the tetrahedron D with vertices (0, 0, 0), (1, 1, 0), (0, 1, 0), and (0, 1, 1).
Solution We sketch D along with its “shadow” R in the xz-plane (Figure 15.29). The
upper (right-hand) bounding surface of D lies in the plane The lower (left-hand)
bounding surface lies in the plane The upper boundary of R is the line
The lower boundary is the line
First we find the y-limits of integration. The line through a typical point (x, z) in R
parallel to the y-axis enters D at and leaves at
Next we find the z-limits of integration. The line L through (x, z) parallel to the z-axis
enters R at and leaves at
Finally we find the x-limits of integration. As L sweeps across R, the value of x varies
from to The integral is
\[\int\limits_{0}^{1}\int\limits_{0}^{1-x}\int\limits_{x+z}^{1}F(x,y,z) dy dz dx\]
=\[\int\limits_{0}^{1}\int\limits_{0}^{1-x}\int\limits_{x+z}^{1} dy dz dx\]
=\[\int\limits_{0}^{1}\int\limits_{x}^{1}(y-x) dy dx\]
=\[\int\limits_{0}^{1}[(1/2)y^2-xy]_{y=x}^{y=1}dx\]

=\[\int\limits_{0}^{1}(1/2-x+(1/2)x^2) dx\]
=\[[(1/2)x-(1/2)x^2+1/6x^3]_{0}^{1}\]
=1/6


Make sure to make a drawing so that you can see where the lines intersect and where it projects onto x plane. Hope this helps.

Answer 2

can you draw a graph? i understand how to integrate for the most part i just dont know how to get the limits for the integrals

Answer 3

The graph is in the attached file.

Answer 4

how would i get the equation from the graph?

Answer 5

the tetrahedron equation?

Answer 6

You need to find the equations of some sides in their planes and the equation of the plane (isoceles triangle)

Answer 7

What is the equation of the plane (isoceles triangle)?

Answer 8

i dont know how to get the equation of the plane

Answer 9

y= x+ z

Answer 10

how did you get y = x+ for the equation of an isosceles triangle?

Answer 11

x+z*

Answer 12

You may have to read about planes and lines and know this subject well. This will help you understand double and triple integration.