Illustrate homework (integral region with implicit region)

Question

I must integrate: $$ \int \int_D x^2y^2 dx dy$$ in the first quadrant.

This is a triangle with vertices in $(0,0)$, $(0,1)$ and $(1,0)$.

I tried to draw it in mathematica with implicit region:

R = ImplicitRegion[{x + y <= 1}, {x, y}]; 
Plot[R, {x, 0, 1}, {y, 0, 1}]

This is wrong and in addition I want to highlight that it is in the first quadrant (highlight $x$ and $y$ axis.

Can someone please help me with that?

Answer 1

Try:

reg = ImplicitRegion[{x + y <= 1, x >= 0, y >= 0}, {x, y}]

You can then visualize the region with RegionPlot[reg], or use it as the domain of an integral:

Integrate[x^2 y^2, Element[{x, y}, reg]]

(* Out: 1/180 *)

Answer 2

But you don't need ImplicitRegion:

Integrate[x^2 y^2, {x, 0, 1}, {y, 0, 1 - x}]

(*

1/180

*)