Respuesta :

Parameterize the intersection by

[tex]\mathbf r(r,\theta)=(r\cos\theta,r\sin\theta,r^2\cos\theta\sin\theta)[/tex]

with [tex]0\le r\le5[/tex] and [tex]0\le\theta\le2\pi[/tex]. Then the area is given by the surface integral

[tex]\displaystyle\int_{r=0}^{r=5}\int_{\theta=0}^{\theta=2\pi}\left\|\mathbf r_r\times\mathbf r_\theta\right\|\,\mathrm d\theta\,\mathrm dr[/tex]
[tex]\displaystyle\int_0^{2\pi}\int_0^5r\sqrt{1+r^2}\,\mathrm dr\,\mathrm d\theta[/tex]
[tex]\displaystyle\pi\int_1^{26}\sqrt s\,\mathrm ds[/tex]
[tex]\displaystyle\frac{2\pi}3s^{3/2}\bigg|_{s=1}^{s=26}=\dfrac{2\pi(26^{3/2}-1)}3[/tex]
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