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I am trying to use the solution $r_0$ of the equation

$r^3 -r b^2 +2b^2=0$

as the limit of the integral

$\int^{r_0}_0 \left[b-b^3 u^3+\frac{b^3 u^2}{2}+\left(\frac{3 b^5}{8}+\frac{2 b^3}{\sqrt{\pi }}\right) u^4 \right] du$

and plot this between $b=5$ and $b=10$.

I tried to use

Plot[Integrate[ b + 1/2 b^3 u^2-b^3 u^3 + (3/8 b^5 + (2  b^3)/Sqrt[\[ Pi]]) 
u^4, {u, 0,  Evaluate[ NSolve[r^3 == b^2 (r - 2), r, Reals]}]]],{b,5,10}]

and

Subscript[r, 0][b_] = NSolve[r^3 == b^2 (r - 2), r, Reals]

Plot[Integrate[b + 1/2  b^3 u^2 - b^3 u^3 + (3/8  b^5 + 
(2b^3)/Sqrt[\[Pi]]) u^4, {u, 0, Subscript[rEvaluate[Subscript[r, 0][b]}]]], {b, 5, 10}]

but these both give me a blank graph

I am trying to use the solution $r_0$ of the equation

$r^3 -r b^2 +2b^2=0$

as the limit of the integral

$\int^{r_0}_0 \left[b-b^3 u^3+\frac{b^3 u^2}{2}+\left(\frac{3 b^5}{8}+\frac{2 b^3}{\sqrt{\pi }}\right) u^4 \right] du$

and plot this between $b=5$ and $b=10$.

I tried to use

Plot[Integrate[ b + 1/2 b^3 u^2-b^3 u^3 + (3/8 b^5 + (2  b^3)/Sqrt[\[ Pi]]) 
u^4, {u, 0,   NSolve[r^3 == b^2 (r - 2), r, Reals]}],{b,5,10}]

and

Subscript[r, 0][b_] = NSolve[r^3 == b^2 (r - 2), r, Reals]

Plot[Integrate[b + 1/2  b^3 u^2 - b^3 u^3 + (3/8  b^5 + 
(2b^3)/Sqrt[\[Pi]]) u^4, {u, 0, Subscript[r, 0][b]}], {b, 5, 10}]

but these both give me a blank graph

I am trying to use the solution $r_0$ of the equation

$r^3 -r b^2 +2b^2=0$

as the limit of the integral

$\int^{r_0}_0 \left[b-b^3 u^3+\frac{b^3 u^2}{2}+\left(\frac{3 b^5}{8}+\frac{2 b^3}{\sqrt{\pi }}\right) u^4 \right] du$

and plot this between $b=5$ and $b=10$.

I tried to use

Plot[Integrate[ b + 1/2 b^3 u^2-b^3 u^3 + (3/8 b^5 + (2  b^3)/Sqrt[\[ Pi]]) 
u^4, {u, 0,  Evaluate[ NSolve[r^3 == b^2 (r - 2), r, Reals]}]],{b,5,10}]

and

Subscript[r, 0][b_] = NSolve[r^3 == b^2 (r - 2), r, Reals]

Plot[Integrate[b + 1/2  b^3 u^2 - b^3 u^3 + (3/8  b^5 + 
(2b^3)/Sqrt[\[Pi]]) u^4, {u, 0, Evaluate[Subscript[r, 0][b]}]], {b, 5, 10}]

but these both give me a blank graph

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Using the solution to an equation as an integral limit (when plotting)

I am trying to use the solution $r_0$ of the equation

$r^3 -r b^2 +2b^2=0$

as the limit of the integral

$\int^{r_0}_0 \left[b-b^3 u^3+\frac{b^3 u^2}{2}+\left(\frac{3 b^5}{8}+\frac{2 b^3}{\sqrt{\pi }}\right) u^4 \right] du$

and plot this between $b=5$ and $b=10$.

I tried to use

Plot[Integrate[ b + 1/2 b^3 u^2-b^3 u^3 + (3/8 b^5 + (2  b^3)/Sqrt[\[ Pi]]) 
u^4, {u, 0,   NSolve[r^3 == b^2 (r - 2), r, Reals]}],{b,5,10}]

and

Subscript[r, 0][b_] = NSolve[r^3 == b^2 (r - 2), r, Reals]

Plot[Integrate[b + 1/2  b^3 u^2 - b^3 u^3 + (3/8  b^5 + 
(2b^3)/Sqrt[\[Pi]]) u^4, {u, 0, Subscript[r, 0][b]}], {b, 5, 10}]

but these both give me a blank graph