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3 years ago

#cancelkeyyon2021 you are the worst mod and I hope you see this

Mathematics

2 answers:

padilas [110]3 years ago

6 0

Answer:

who even is the mod?! for what service?!

8090 [49]3 years ago

4 0

Answer:

who is that?

Step-by-step explanation:

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Find, correct to four decimal places, the length of the curve of intersection of the cylinder 16x2 + y2 = 16 and the plane x + y

lubasha [3.4K]

First find a parameterization for the curve of intersection.

Given the equation of a cylinder, a natural choice for a parameterization would be one utilizing cylindrical coordinates. Here,

$16x^2+y^2=16\implies x^2+\left(\dfrac y4\right)^2=1$

which suggests we could use

$\begin{cases}x(t)=\cos t\\y(t)=4\sin t\\z(t)\end{cases}$

with $0\le t\le2\pi$ , and we get $z(t)$ from the equation of the plane,

$x+y+z=12\implies z(t)=12-x(t)-y(t)=12-\cos t-4\sin t$

Now use the arc length formula:

$\displaystyle\ell=\int_0^{2\pi}\sqrt{\left(\dfrac{\mathrm dx}{\mathrm dt}\right)^2+\left(\dfrac{\mathrm dy}{\mathrm dt}\right)^2+\left(\dfrac{\mathrm dz}{\mathrm dt}\right)^2}\,\mathrm dt$

$\displaystyle\ell=\int_0^{2\pi}\sqrt{\sin^2t+16\cos^2t+(\sin t-4\cos t)^2}\,\mathrm dt$

$\displaystyle\ell=\sqrt2\int_0^{2\pi}\sqrt{\sin^2t-4\cos t\sin t+16\cos^2t}\,\mathrm dt\approx\boxed{24.0878}$

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4 years ago