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The mass of a thin plate M is given below in the conditional units:

$M=\int_0^1 [\int_0^x (x+y^2)dy]dx+ \int_1^2 [\int_0^{2-x}(x+y^2)dy]dx$

Determine the function describing the surface mass desity...

I'm not too sure how to tackle this problem. If I solve the double integral I get ${5\over 12}+{3\over 4}$ but I'm lost at trying to find the function. Is this a Double Riemann Sums problem?

Update: So the region from the first integral is $0\le x \le 1, 0\le y \le x$ and the second integral region is $1\le x \le 2, 0\le y \le 2-x $

I'm not seeing how these boundary conditions are interpret with vertices at (0,0), (1,1) and (2,0)

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I'm not sure if I completely understand your question. For a thin plate with a shape described by the region $R$ in the $xy$-plane and with mass density $m(x,y)$, the mass $M$ is given by: $$M = \iint_R m(x,y) \, dxdy$$ From the integrals giving the mass in your case, you can derive that the thin plate is a triangle (interpret the integral boundaries, make a sketch) with vertices at $(0,0)$, $(1,1)$ and $(2,0)$ and with mass density $m(x,y) = x+y^2$.

So to answer your question, I think you just have to 'read' the function being integrated and by inspection, you find the mass density function as the integrand: $m(x,y) = x+y^2$.

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