The partial of (Subscript[F, 2]) with respect to x is

3(x^2+y^2)

Thus (Subscript[F, 2]) is

Integrate[3(x^2+y^2),x]

partial wrt y of (Subscript[F, 1]) is

(((x^2+y^2))^(3/2))

Thus (Subscript[F, 1]) is

FullSimplify[Integrate[(((x^2+y^2))^(3/2)),y]]

{(1/8) (y (Sqrt[(x^2)+(y^2)]) (5 (x^2)+2 (y^2))+3 (x^4) Log[y+(Sqrt[(x^2)+(y^2)])]),(x^3)+3 x (y^2)}

zero work

Integrate[Integrate[(3 (r^2)-(r^3)) r,{r,0,3}],{θ,0,2 Pi}]

Integrate[Integrate[(3 (r^2)-(r^3)) r,{r,0,a}],{θ,0,2 Pi}]

Solve[2 (((3 (a^4))/4)-((a^5)/5)) π==0,a]

Plot[2 (((3 (a^4))/4)-((a^5)/5)),{a,0,4}]

D[2 (((3 (a^4))/4)-((a^5)/5)),a]

Solve[2 (3 (a^3)-(a^4))==0,a]

without Green’s thm

{(1/8) (y (Sqrt[(x^2)+(y^2)]) (5 (x^2)+2 (y^2))+3 (x^4) Log[y+(Sqrt[(x^2)+(y^2)])]),(x^3)+3 x (y^2)}

{a Cos[t],a Sin[t]}

D[{a Cos[t],a Sin[t]},t]

Feed the curve into the v. field

FullSimplify[{(1/8) (y (Sqrt[(x^2)+(y^2)]) (5 (x^2)+2 (y^2))+3 (x^4) Log[y+(Sqrt[(x^2)+(y^2)])]),(x^3)+3 x (y^2)}/.{x-> a Cos[t],y-> a Sin[t]},a>0]

FullSimplify[{(1/8) (a^4) (3 ((Cos[t])^4) Log[a (1+Sin[t])]+5 ((Cos[t])^2) Sin[t]+2 ((Sin[t])^3)),-(a^3) Cos[t] (-2+Cos[2 t])}.{-a Sin[t],a Cos[t]}]

Integrate[-(1/8) (a^4) (3 a ((Cos[t])^4) Log[a (1+Sin[t])] Sin[t]+2 a ((Sin[t])^4)+((Cos[t])^2) (-16+8 Cos[2 t]+5 a ((Sin[t])^2))),{t,0,2 Pi}]

Plot[(1/10) (15-4 a) (a^4) π,{a,0,4}]

Problem 28

cp=ContourPlot[(x^3)+(y^3)-3 x y,{x,0,2},{y,0,2},Contours->{0}]

pp=ParametricPlot[{((3 t)/(1+(t^3))),((3 (t^2))/(1+(t^3)))},{t,0,100},PlotRange->{{0,2},{0,2}},PlotStyle-> {Red,Thickness[0.02]}]

Show[cp,pp]

FullSimplify[((x^3)+(y^3)-3 x y)/.{x-> ((3 t)/(1+(t^3))),y-> ((3 (t^2))/(1+(t^3)))}]

D[{((3 t)/(1+(t^3))),((3 (t^2))/(1+(t^3)))},t]

Feed the curve to the v. field

{0,x}/.{x-> ((3 t)/(1+(t^3))),y-> ((3 (t^2))/(1+(t^3)))}

Dot it

FullSimplify[{0,((3 t)/(1+(t^3)))}.{-((9 (t^3))/(((1+(t^3)))^2))+(3/(1+(t^3))),-((9 (t^4))/(((1+(t^3)))^2))+((6 t)/(1+(t^3)))}]

Integrate[-((9 (t^2) (-2+(t^3)))/(((1+(t^3)))^3)),{t,0,Infinity}]