Respuesta :

LammettHash
Parameterize the surface (call it [tex]\mathcal S[/tex]) by

[tex]\mathbf s(u,v)=\langle u\cos v,u\sin v,u^2\sin^2v-u^2\cos^2v\rangle=\langle u\cos v,u\sin v,-u^2\cos2v\rangle[/tex]

with [tex]4\le u\le5[/tex] and [tex]0\le v\le2\pi[/tex], which yields

[tex]\mathrm dS=\|\mathbf s_u\times\mathbf s_v\|\,\mathrm du\,\mathrm dv=u\sqrt{1+4u^2}\,\mathrm du\,\mathrm dv[/tex]

Then the area of [tex]\mathcal S[/tex] is given by the surface integral

[tex]\displaystyle\iint_{\mathcal S}\mathrm dS=\int_{v=0}^{v=2\pi}\int_{u=4}^{u=5}u\sqrt{1+4u^2}\,\mathrm du\,\mathrm dv=\dfrac{(101^{3/2}-65^{3/2})\pi}6[/tex]
nutanraj654

The part of the hyperbolic paraboloid z = y2 − x2 that lies between the cylinders x2 + y2 = 16 and x2 + y2 = 25.

What is the formula for a hyperbolic paraboloid?

The basic hyperbolic paraboloid is given by the equation z=Ax2+By2 z = A x 2 + B y 2 where A and B have opposite signs.

What is hyperbolic paraboloid?

Definition of a hyperbolic paraboloid

: a saddle-shaped quadric surface whose sections by planes parallel to one coordinate plane are hyperbolas while those sections by planes parallel to the other two are parabolas if the proper orientation of the coordinate axes is assumed.

Learn more about the part of the hyperbolic paraboloid at

https://brainly.com/question/6769553

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