1. ## Help please in evaluating Triple Integral

I need help in starting this question, How do I draw the region R and how do I find the limits?

Thanks in advance for any help.

Below is my (very rough!) interpretation of the region R, the red lines represent the two major planes.

2. You have that $0 \leq z \leq x$ and $x^3 \leq y \leq 8$

That gives us the x range $0 \leq x$ and $x^3 \leq 8 \Longleftrightarrow x \leq 2$

3. Originally Posted by dgmossman
I need help in starting this question, How do I draw the region R and how do I find the limits?
Below is the Mathematica code to draw the region. So it's that wedge in there right? Can you look at the code and determine what each curve (blue, light purple, and light red) are?

So if the integral is:

$\int\int\int f dzdydx$

can you look at the plot and determine first what the limits on z are, then the limits on y, then finally x?

Code:
poly1 = Graphics3D[Polygon[{{4, 8, 0}, {-4, 8, 0},
{-4, 0, 0}, {4, 0, 0}}]];
cp1 = ContourPlot3D[y == x^3, {x, 0, 2}, {y, 0, 8},
{z, 0, 4}, ContourStyle -> {Opacity[0.4], LightPurple},
Mesh -> None]
cp2 = ContourPlot3D[z == x, {x, 0, 4}, {y, 0, 8},
{z, 0, 5}, ContourStyle -> {Opacity[0.4], LightGreen},
Mesh -> None]
Show[{poly1, cp1, cp2}, PlotRange ->
{{0, 4}, {0, 8}, {0, 5}}, Axes -> True,
AxesLabel -> {Style["X", 10], Style["Y", 10],
Style["Z", 10]}, AxesEdge -> {{-1, -1}, {1, -1},
{-1, -1}}, BoxRatios -> {1, 1, 1}]

4. Thanks for that, but-I was nearly there... But how do you set it up so that all of the variables are eliminated. Obviously U need to integrate w.r.t. x last, but what order do I integrate the other two variables?

5. It doesn't really matter, although I would suggest you start with z first.

6. Thanks very much guys! Glad there is people like you out there to help.

7. Originally Posted by dgmossman
Thanks for that, but-I was nearly there... But how do you set it up so that all of the variables are eliminated. Obviously U need to integrate w.r.t. x last, but what order do I integrate the other two variables?
Alright, look at the plot, z is going from the bottom surface (z=0) to the top surface of the wedge. But that surface is just z=x. Now look at how we'd integrate over y: It's going from that curve in the x-y plane (y=x^3) to the back side you said is y=8. How about x? It's going from 0 to the point where 8=x^3. Can you see how I'd write:

$V=\int_0^2 \int_{x^3}^8 \int_0^x x\,dzdydx$

How about since I did that one to to it this way then:

$\int\int\int x\,dzdxdy$

8. Thanks shawsend, your explanation was great- I understand it now.