Solving a definite integral with constraints on the limits

Question

I have the following definite double integral.

$\int _0^{\frac{d}{1-r}}\int _{\frac{d+r x-x}{r}}^s [(1-r) x+r y -d]dydx$

with the constraints: $0\leq d \leq 1$, $1 \leq s \leq 4$, $0 \leq r \leq 1$, $0 \leq x \leq s$, $0\leq y \leq s$.

My code for solving this integration is

Integrate[(1 - r)*x + r*y - d, {x, 0, d/(1 - r)}, {y, (d + r*x - x)/r,s}]

And I got

(d^3 - 3 d^2 r s + 3 d r^2 s^2)/(6 r - 6 r^2)

But I think this is not correct because the variables $x$ and $y$ are bounded by $0$ and $s$ and therefore I think I need additional constraints on the limits, i.e.

$\frac{d}{1-r}\leq s$ and $0 \leq \frac{d+rx-x}{r} \leq s$.

And I'm wondering how to add these two constraints in solving the above integration. Can anyone help please? Thanks!

Answer 1

You have to modify the integration limits to {x, 0, Min[s, d/(1 - r)]}, {y, Max[0, Min[s, (d + r*x - x)/r]], s}

Because Mathematica can't solve symbolica I give a numeric solution

int[s_?NumericQ, d_?NumericQ, r_?NumericQ] :=NIntegrate[(1 - r)*x + r*y - d, {x, 0, Min[s, d/(1 - r)]}, {y,Max[0, Min[s, (d + r*x - x)/r]], s} ]

The result can be plotted for variing s in the parameterrange 0<d,r<1

Plot3D[ int[#, d, r], {d, 0, 1}, {r, 0, 1}, AxesLabel -> {d, r, None},PlotLabel -> "s== " <> ToString[#]] &[2]