Describe the region bounded by the planes: x=0, y=0, z=0, x+y=4, and x=z-y-1.

It just says to describe the region. 2 things.

  1. does anyone know any software that will allow me to draw out these regions.
  2. how do i describe this region?

(see below a picture, done with Matlab).

The 3 first equations show the shape is in the first orthant.

This shape can easily be pictured as a house whose unique room has a right isosceles floor on the ground $z=0$, on which are raised vertical walls :

  • $W_1 \perp W_2$ with equations $x=0, \ y=0$ resp. and

  • $W_3$ with equation $x+y=4$ making a $45$° angle with $W_1$ and $W_2$.

The last equation describes a roof beginning at height $1$ meter (say) above the origin, reaching five meters above wall $W_3$ because $z=x+y+1$ takes uniform value $5$ when $x+y=4$.

Remark: The slope of the roof is :

$$\text{in radians :} \ \ \arctan(\sqrt{2}) \ \ \approx \ \ 54.7°.$$

enter image description here

  • $\begingroup$ a graph would really help, but how do i find the bounds of integration when just given the planes. i'm having a lot of difficulty visualizing the different planes in the x,y,z plane and help would be really appreciated. $\endgroup$ – Math19384 Apr 7 '18 at 22:07
  • $\begingroup$ Graphical representation added. $\endgroup$ – Jean Marie Apr 7 '18 at 23:03
  • $\begingroup$ I think that you have no difficulty for the first 3 planes. About the fourth, you begin by drawing straight line with equation $y=-x+4$ in the horizontal plane ; as there are no constraint on $z$ any point at the vertical of this line is convenient : more generally, any equation $y=ax+b$ considered in 3D space is the equation of a vertical plane. Lastly, the fifth plane P has equation $z=x+y+1$. If you are looking where P intersects the $z$ axis, you make $x=y=0$ in its equation and you get $z=1$. Same thing for the point at the vertical of $(x=4,y=0)$. This time, you get $z=4+0+1=5$. $\endgroup$ – Jean Marie Apr 7 '18 at 23:13

(1) If you are using a Mac, then the application Grapher is really helpful and easy to use (I believe it comes with your Mac, not sure).

(2) There is no unique way to describe it. You can say, for example:

$$ S= (x,y,z) \in \Bbb R^3|x>0 \; , \; y>0 \; , \; z>0 \; , \; z-x-1<y<4-x$$


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