# Evaluate the surface integrals using divergence theorem ${\oint \oint}_S (xy\bar{i} + z^2 \bar{k}) \bar{n} dS$. Help finding domain

Evaluate the surface integrals using divergence theorem

$${\oint \oint}_S (xy\bar{i} + z^2 \bar{k}) \bar{n} dS$$

where S is the surface enclosing the volume in the first octant bounded by the planes $$z= 0, y = x, y = 2x, x + y+ z = 6$$ and $$\bar{n}$$ is the unit outer normal to S.

Ans:

So by divergence theorem:

$${\oint \oint}_S (xy\bar{i} + z^2 \bar{k}) \bar{n} dS = \int \int_{\Omega} (y + 2z) dV$$

now adding the respective integral limits

$$\int_{0}^{2} \int_{x}^{2x} \int_{0}^{6 - x - y} (y - 2z) dz dy dx + \int_{2}^{3} \int_{x}^{6-x} \int_{0}^{6 - x - y} (y - 2z) dz dy dx$$

so the domain for each variable is

$$0 \leq z \leq 6 -x - y$$ they just rearrange the equation with respect to $$z$$

$$x \leq y \leq 2x$$ (this was given in the question)

$$0 \leq x \leq 2$$ (I don't understand how they got this could someone explain. Thank you.)

for the second integral:

$$0 \leq z \leq 6 -x - y$$ they just rearrange the equation with respect to $$z$$

$$x \leq y \leq 6-x$$ since $$z$$ is done with we can consider $$z = 0$$ and solve for $$y$$.

$$2 \leq x \leq 3$$ (I don't understand this either)

Not trying to solve the integral just wondering how they got the domain

## 1 Answer

The volume of integration is delimited by vertical planes and only two non vertical ones, this makes things very clear for the $$z$$ part of the integral (as indeed you checked). So, we need to understand the limits for $$x$$ and $$y$$ into the $$z=0$$ plane. These are determined by the intersection of three planes with the plane $$z=0$$, so is, these three lines

$$y=x$$, $$y=2x$$ and $$x+y=6$$. They define a triangle with vertex at $$(0,0)$$, $$(3,3)$$ and $$(2,4)$$

Now, we have to describe de range for the variation of $$y$$ depending on the value of $$x$$ we are considering.

If $$0\leq x\leq 2$$ then $$x\leq y\leq 2x$$

If $$2< x\leq 3$$ then $$x\leq y\leq 6-x$$

A picture helps to visualize. 