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tatiyna
3 years ago
12

Let c be the curve of intersection of the parabolic cylinder x2 = 2y, and the surface 3z = xy. find the exact length of c from t

he origin to the point (2, 2, 4/3).
Mathematics
1 answer:
Mandarinka [93]3 years ago
3 0
Parameterize the intersection by setting x(t)=t, so that

x^2=2y\iff y=\dfrac{x^2}2\implies y(t)=\dfrac{t^2}2
3z=xy\iff z=\dfrac{xy}3\implies z(t)=\dfrac{t^3}6

The length of the path C is then given by the line integral along C,

\displaystyle\int_C\mathrm dS

where \mathrm dS=\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. We have

\dfrac{\mathrm dx}{\mathrm dt}=1
\dfrac{\mathrm dy}{\mathrm dt}=t
\dfrac{\mathrm dz}{\mathrm dt}=\dfrac{t^2}2

and so the line integral is

\displaystyle\int_{t=0}^{t=2}\sqrt{1^2+t^2+\dfrac{t^4}4}\,\mathrm dt

This result is fortuitous, since we can write

1+t^2+\dfrac{t^4}4=\dfrac14(t^4+4t^2+4)=\dfrac{(t^2+2)^2}4=\left(\dfrac{t^2+2}2\right)^2

and so the integral reduces to

\displaystyle\int_{t=0}^{t=2}\frac{t^2+2}2\,\mathrm dt=\dfrac{10}3
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