Answer to Question #285587 in Analytic Geometry for Nikki

Question #285587

Find the length of the arc of the cal x square is equal to eat from the vertex to an activity of a letter rectum

1
Expert's answer
2022-01-11T13:46:37-0500

Find length of the arc of the parabola x2=4ayx^2=4ay measured from the vertex to one extremity of the Latus-Rectum

Latus-Rectum is a chord that passes through a focus and is parallel to the directrix


Let A be the vertex and L an extremity of the Latus-Rectum so that at A,x=0 and at L,x=2a.

Then:

y=x2/(4a),y=x/(2a)y=x^2/(4a),y'=x/(2a)

focus: (a,0)(-a,0)

directrix: x=ax=a


arc AL=02a1+(y)2dx=02a1+(x2a)2dx=AL=\int^{2a}_0\sqrt{1+(y')^2}dx=\int^{2a}_0\sqrt{1+(\frac{x}{2a})^2}dx=


x=2asinhudx=2acoshudux=2asinhu\to dx=2acoshudu


=2acoshu4a2sinh2u+4a2du=4a2cosh2udu==\int 2acoshu\sqrt{4a^2sinh^2u+4a^2}du=4a^2\int cosh^2udu=


=coshusinhu+u2=(x(x/(2a))2+12+sinh1x2a)02a=a(2+sinh11)=\frac{coshusinhu+u}{2}=(\frac{x\sqrt{(x/(2a))^2+1}}{2}+sinh^{-1}\frac{x}{2a})|^{2a}_0=a(\sqrt 2+sinh^{-1}1)


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