Welcome to an example of how to set up
the limits of integration for a triple integral.
Here we're given the region T as the tetrahedron
with the vertices shown here, where we have an x intercept,
a y intercept, a z intercept and the origin.
So this tetrahedron is in the first octant.
We're asked to write the triple integral in this form here,
where notice how the order of integration
is first with respect to x, then with respect to y,
then with respect to z.
Let's first take a look at the region T in space.
The region of integration is this tetrahedron here,
where we have four vertices.
We have this x intercept, this y intercept,
this z intercept, and the origin.
So notice how the tetrahedron is bounded
by the three corner planes and this green plane.
So our first step is going to be to find
the equation of this green plane.
For review, if we have the x, y and z intercepts of a plane,
we can write the equation of the plane in this form here.
So in our case, the plane is going to have the equation
x divided by three
plus y divided by two
plus z divided by four equals one.
Now from here let's go ahead and clear the fractions
by multiplying through by the least common denominator,
which is the least common multiple
of three, two and four, which is 12.
So multiply through by 12.
We can simplify before multiplying here on the left.
There's one three in three and four threes in 12,
so we have four x plus.
There's one two in two and six twos in 12, so plus six y.
Plus, there's one four in four and three fours in 12,
so plus three z equals 12.
Which means the region T, the tetrahedron, is bounded
by the three coordinate planes which would be x equals zero,
y equals zero, z equals zero, and this plane here,
four x plus six y
plus three z equals 12.
Now going back to our triple integral for a moment,
we first integrate with respect to x.
So we're going to integrate from zero to the plane
where the equation of the plane must be expressed
as a function of y and z,
which means we must solve the equation of the plane for x.
So solving this equation for x,
we would subtract six y on both sides
as well as subtract three z on both sides.
So it would give us four x equals 12
minus six y minus three z.
Divide everything by four,
and we get x equals 3 minus
6/4 simplifies to 3/2,
so 3/2 y minus 3/4 z.
Which means limits of integration, with respect to x
are going to be from zero,
to v of y comma z,
which is really just x
equals three minus 3/2 y minus 3/4 z.
So this is the first part of our question.
And now we still have to integrate with respect to y
and then z, which means we now want to find the y z trace.
We can find the y z trace by setting x equal to zero.
So the y z trace
is going to be the region bounded by y equals zero,
z equals zero, and this plane here when x equals zero.
Which would give us the equation
six y plus three z equals 12.
So to graph the y z trace you want to graph
these three equations on the y z plane
to determine the bounded region.
And Bri had done this to save some time.
Where here's the y z trace,
where here's the y axis and here's the z axis.
So this side of the triangular region
would be z equals zero.
This side of the triangular region would be y equals zero.
And this side of the triangular region
would be on the line,
six y plus three z equals 12.
Now we can determine the limits of integration for y.
For y we're going to integrate from y equals zero
to this line here,
which must be expressed as a function of z,
which means we need to solve this equation here for y.
To solve this equation for y, we would subtract three z
on both sides.
That would give us six y equals 12 minus three z.
We divide both sides by six,
and so we'd have y equals two minus 1/2 z,
which is the upper limit of integration for y.
And for our problem we're asked to express it as u of z.
So u of z, which is really just y,
is equal to two minus 1/2 z.
And then for limits of integration for z
which are already given, they're from zero to four.
And that's because looking at the y z trace,
notice how we integrate from z equals zero
up to this high point where z equals four.
I hope you found this helpful.