# How to solve and plot Definite Integral with Symbolic Limits

I need to plot g1, g2, g3 and g4 expressions. To do that, I am trying to evaluate the double defined integral in the three expressions, however I am having issues because one of the integration limits (alpha) is function of w. The resultante error says:

NIntegrate::nlim: α = α is not a valid limit of integration.

The question here is how can I compute a definite integral if I have a symbolic limit? Is it possible to do that?

Otherwise, is there anothet way to calculate and plot these expressions solving the indefinite integral?

d = 0.0254;
young = 210 10^9;
iner = (Pi/64)*d^4;
ld = 24;
L = ld*d;
wcord = Sqrt[a*(d - a)];

α = a - (d/2) + Sqrt[((d/2)^2) - (w^2)];
αp = Sqrt[d^2 - (2*w)^2];
rα = α/αp;
F = Sqrt[(2*αp)/(Pi*α)*Tan[(Pi*α)/(2*αp)]]*((0.923 + 0.199*(1 - Sin[(Pi*[Alpha])/(2*αp)])^4)/Cos[(Pi*α)/(2*αp)]);

Fp = Sqrt[(2*αp)/(Pi*α)*Tan[(Pi*α)/(2*αp)]]*((0.752 + 2.02*rα + 0.37*(1 - Sin[(Pi*α)/(2*αp)])^3)/Cos[(Pi*α)/(2*αp)]);

flxbInt = L^3/(48*young*iner) ;

g1 = flxbInt + 2*(NIntegrate[((128*(L^2)*(αp^2))/(young*Pi*(d^8)))*α*F^2, {α, 0, α}, {w, 0, wcord}])

g4 = flxbInt + 2*(NIntegrate[((512*(L^2)*(w^2))/(young*Pi*(d^8)))*α*Fp^2, {α, 0, α}, {w, 0, wcord}])

g2 = g3 = 2*(NIntegrate[((256*(L^2)*αp*w)/(young*Pi*(d^8)))*α*F*Fp, {α, 0, α}, {w, 0, wcord}]) • There is a type in the definition of F; also, change the dummy integration variable. Oct 28 '16 at 7:24
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– user9660
Oct 28 '16 at 8:39

The most direct way of tackling this problem is defining a unnesting the integrals, converting to functions, and using NIntegrate

As before:

d = 0.0254;
young = 210 10^9;
iner = (Pi/64)*d^4;
ld = 24;
L = ld*d;
wcord = Sqrt[a*(d - a)];

\[Alpha] = a - (d/2) + Sqrt[((d/2)^2) - (w^2)];
\[Alpha]p = Sqrt[d^2 - (2*w)^2];
r\[Alpha] = \[Alpha]/\[Alpha]p;

F = Sqrt[(2*\[Alpha]p)/(Pi*\[Alpha])*Tan[(Pi*\[Alpha])/(2*\[Alpha]p)]]*((0.923 + 0.199*(1 - Sin[(Pi*\[Alpha])/(2*\[Alpha]p)])^4)/Cos[(Pi*\[Alpha])/(2*\[Alpha]p)]);
Fp = Sqrt[(2*\[Alpha]p)/(Pi*\[Alpha])*Tan[(Pi*\[Alpha])/(2*\[Alpha]p)]]*((0.752 + 2.02*r\[Alpha] + 0.37*(1 - Sin[(Pi*\[Alpha])/(2*\[Alpha]p)])^3)/Cos[(Pi*\[Alpha])/(2*\[Alpha]p)]);
flxbInt = L^3/(48*young*iner);


Now define

i1[z_?NumericQ] := NIntegrate[k, {k, 0, z}];


Rewriting the integrals in terms of our new function i1

g1 = flxbInt + 2*(NIntegrate[((128*(L^2)*(\[Alpha]p^2))/(young*Pi*(d^8)))*i1[\[Alpha]]*F^2, {w, 0, wcord}])
g4 = flxbInt + 2*(NIntegrate[((512*(L^2)*(w^2))/(young*Pi*(d^8)))*i1[\[Alpha]]*Fp^2, {w, 0, wcord}])
g2 = g3 = 2*(NIntegrate[((256*(L^2)*\[Alpha]p*w)/(young*Pi*(d^8)))*i1[\[Alpha]]*F*Fp, {w, 0, wcord}])


Evaluating the integrals yields:

(*g1 = 1.20937*10^-6*)
(*g4 = 1.13447*10^-6*)
(*g2 = g3 = 4.88326*10^-8*)