September 11th, 2017, 09:05 AM  #1 
Senior Member Joined: Nov 2015 From: United States of America Posts: 165 Thanks: 21 Math Focus: Calculus and Physics  Graphing a triple integral
Hello forum! I am trying to graph this triple integral. $$\int\limits_{1}^{1}\int\limits_{y}^{(y+3)}\int\limits_{(x+2)}^{ (x+4)} f(x,y,z) \,dz\,dx\,dy$$ I would present my attempt, but it simply looks like a 3D coordinate systems with a bunch of lines drawn in places that I believe might be right, but I just can't draw 3D representations well and my professor expects a sketch. I am thinking about trying to throw this into mathematica and then just sketching that. I began with restricting the y domain to $1 \leq \, y \leq \, 1$. Then I focused on the xy plane only. I have two functions of interest for the xy plane. These are $x(y) = y+3$ and $x(y) = y$. I came to the conclusion that the projection of this 3D object onto the xyplane is a parallelogram. I then got two more functions to investigate. $z(x) = x+2$ and $z(x) = x+4$. These were a set of two lines. So my object doesn't even touch the xy plane? It is a good bit above the xy plane? Am I thinking about this correctly? Are there better methods / thought processes to solve these problems? Thanks! Jacob 
September 11th, 2017, 09:28 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,790 Thanks: 922 
Suppose you have a planar figure given by $(f(x),f(y))$ adding a zcoordinate that is equal to $f(x)$ has the effect of rotating the planar figure about the $y$ axis $45^\circ$ changing that zcoordinate to $f(x)+z_0$ adds a displacement in the $z$ direction. So what's going on is that your parallelogram in the $xy$ plane is simultaneously rotated by $45^\circ$ and translated up by $2$ and $4$ units. Then all the corners are connected by vertical lines and the 3D integration area is that tilted parallelogram prism. You are correct that it is above the $xy$ plane. 
September 11th, 2017, 09:59 AM  #3 
Senior Member Joined: Nov 2015 From: United States of America Posts: 165 Thanks: 21 Math Focus: Calculus and Physics  

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graphing, integral, triple 
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