Respuesta :

LammettHash
Parameterizing [tex]\mathcal C[/tex] by

[tex]\mathbf r(t)=(t^3,t)[/tex]

with [tex]0\le t\le2[/tex], we have

[tex]\mathrm ds=\|\mathbf r'(t)\|\,\mathrm dt=\sqrt{(3t^2)^2+1^2}\,\mathrm dt=\sqrt{9t^4+1}\,\mathrm dt[/tex]

So

[tex]\displaystyle\int_{\mathcal C}y^3\,\mathrm ds=\int_{t=0}^{t=2}t^3\sqrt{9t^4+1}\,\mathrm dt[/tex]
[tex]=\displaystyle\frac1{36}\int_0^236t^3\sqrt{9t^4+1}\,\mathrm dt[/tex]
[tex]=\displaystyle\frac1{36}\int_{u=1}^{u=145}\sqrt u\,\mathrm du[/tex]
[tex]=\dfrac{145^{3/2}-1}{54}[/tex]
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