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Find, correct to four decimal places, the length of the curve of intersection of the cylinder 4x2 + y2 = 4 and the plane x + y + z = 9.

Respuesta :

LammettHash
[tex]4x^2+y^2=4\iff x^2+\left(\dfrac y2\right)^2=1[/tex]
[tex]x+y+z=9\iff z=9-x-y[/tex]

Let [tex]x=\cos t[/tex] and [tex]y=2\sin t[/tex]. Then the cylinder is given when we take [tex]0\le t\le2\pi[/tex].

The cylinder's equation is independent of [tex]z[/tex], which means the intersection is completely determined by the value of [tex]z[/tex] in the plane equation. So the intersection can be described by the vector function [tex]\mathbf r(t)=\langle\cos t,2\sin t,9-\cos t-2\sin t\rangle[/tex].

The length of the intersection curve [tex]C[/tex] is given by the line integral

[tex]\displaystyle\int_C\mathrm d\mathbf r=\int_{t=0}^{t=2\pi}\|\mathbf r'(t)\|\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^{2\pi}\sqrt{\cos^2t+4\sin^2t+(9-\cos t-2\sin t)^2}\,\mathrm dt[/tex]
[tex]\approx57.4449[/tex]

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