Graphing region of integration of triple integral ∫01∫01-x^2∫01-x f(x,y,z)dzdydx

[Lavender786]

Graphing region of integration of triple integral ∫01∫01-x^2∫01-x f(x,y,z)dzdydx

We have to consider the integral of

[FONT=MathJax_Size2]∫₀¹[/FONT][FONT=MathJax_Size2]∫[/FONT]₀^{1-x^2}[FONT=MathJax_Size2]∫₀^1-x [/FONT][FONT=MathJax_Math-italic]f[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]z[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Math-italic]d[/FONT][FONT=MathJax_Math-italic]z[/FONT][FONT=MathJax_Math-italic]d[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Math-italic]d[/FONT][FONT=MathJax_Math-italic]x[/FONT]​

First we're told to graph the region of integration, so I started that but I'm not exactly sure what I'm doing. I end up with something that looks like a slim triangular slice of a cone, but I don't know if it's correct. Is it? Please help

We also have to write the integral in 5 other ways and I'm not sure how.

 

[tkhunny]

I can't help you with the drawing, but you have...

z = 1 - x -- That's a nice lean-to.
y = 1 - x^2 -- That's a nice parabolic cap.

You have everything starting at 0, so we're in the first octant.

Your integral is dzdydx, the five other ways are:

dzdxdy
dydzdx
dydxdz
dxdydz
dxdzdy

Your only stopping point should be x = f(y). Make sure you pick the positive branch.