Find length of the arc of the parabola x2=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)
focus: (−a,0)
directrix: x=a
arc AL=∫02a1+(y′)2dx=∫02a1+(2ax)2dx=
x=2asinhu→dx=2acoshudu
=∫2acoshu4a2sinh2u+4a2du=4a2∫cosh2udu=
=2coshusinhu+u=(2x(x/(2a))2+1+sinh−12ax)∣02a=a(2+sinh−11)
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