# Triple Integral

Given the triple integral $$\int\int\int_{G}$$ xyz dV
Where G is the region bounded by x=1, y=x, y=0, z=0, z=2.
How do I evaluate it.

daniel_i_l
Gold Member
First of all, is there any other constraint on x? I'll assume that x>=0.
Do you know how to change triple integrals into single-variable integrals?
The general idea is to first evaluate the integral while pretending that 2 of the variables are constant. Then you use that result to integrate over the other variables.

In this case it might be easiest to first evaluate the integral in the triangle 0<=x<=1 and
0<=y<=x while assuming that z is constant. Then integrate that result with z as a variable from 0 to 2. The triangle can be evaluated in a similar way. In other words:
$$I = \int^{2}_{0}(\int^{1}_{0}(\int^{1-x}_{0} xyz dy)dx)dz$$

Cheers. I got an answer of 1/3