how do I solve a double integral in Mathematica? -


i new mathematica, , trying solve following problem.

  1. i have cubic equation of form z = az^3 + bz^2 + + b. first thing want function solves analytically z , chooses minimal positive root that, function of , b.

i thought in order root use:

z = solve[z == az^3 + bz^2 + + b, z];

it seems not quite getting roots, expect using general cubic equation solution formula.

  1. i want integrate minimal positive root of z on a , b (again, preferably analytically) 0 1 a , a 1 b.

i tried

y = integrate[z, {a, 0, 1}, {b, a, 1}];

and not seem give formula or numerical value, returns integral. (notice not sure how pick minimal positive root, playing around mathematica try figure out.)

any ideas on how this?

spaces between a or b , z important. can roots by:

sol = z /. solve[z == z^3 + b z^2 + + b, z]

however, sure expression has solution expect? a=0.5 , b=0.5, real root negative.

sol /. {a->0.5, b->0.5} {-2.26953,0.634765-0.691601 i,0.634765+0.691601 i}

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