Question 1041089
You have a thin plate in the first quadrant of the x, y-plane. Its boundaries are +y+=+x%5E2+ and +x+=+y%5E2+, and
its density is ρ(x, y) = +%28x+%2B+1%29%28y+%2B+1%29+ g/cm^2 
<pre>
That's too much to ask.  I'll do the first.

(a) Set up and calculate a double integral for its area A.

{{{drawing(300,300,-.2,1.2,-.2,1.2,
graph(300,300,-.2,1.2,-.2,1.2,sqrt(x)),
graph(300,300,-.2,1.2,-.2,1.2,x^2),
locate(1,1,"(1,1)"), rectangle(.4,.4^2,.42,sqrt(.42)),
locate(.44,.4^2+.1,y=x^2),locate(.44,sqrt(.42)+.04,y=sqrt(x))


 )}}}

{{{int(""^""^""^"",""^""^""^"",0^"",1^"")}}}{{{int(matrix(1,2,"dy"^""^""^"",""),dx^""^""^"",x^2,sqrt(x))}}}{{{""=""}}}{{{int(""^""^""^"","",0,1)}}}{{{(int(matrix(1,2,"dy"^""^""^"",""),"",x^2,sqrt(x)))dx^""^""^""}}}{{{""=""}}}{{{int((matrix(3,3,"","|",sqrt(x),y,"|","","","|",x^2)),dx,0,1)}}}{{{""=""}}}{{{int((sqrt(x)^""-x^2 )^""^"",dx,0,1)}}}{{{""=""}}}{{{int((x^(1/2)-x^2 )^""^"",dx,0,1)}}}{{{""=""}}}

{{{matrix(3,3,
"","|",1,
(x^(3/2)/expr(3/2)^""-x^3^""/3^""),"|","",
"","|",0)

}}}{{{""=""}}}{{{matrix(3,3,
"","|",1,
( expr(2/3)x^(3/2)-x^3/3),"|","",
"","|",0)}}}{{{""=""}}}{{{

matrix(2,1,"",

(expr(2/3)*1^(3/2)-1^3/3)-(expr(2/3)*0^(3/2)-0^3/3)) )}}}{{{""=""}}}{{{(2/3-1/3)-(0-0)}}}{{{""=""}}}{{{1/3}}}

That's enough. The rest is a matter of setting up similar double 
integrals using the formulas in your book for moments, mass, 
average density, and moments of inertia.  Look at examples of
these in your book.  You can do it!

Edwin</pre>