Area of a Conic Arc (Parabola)

caarea2.html

Exploration

Show that the area between a conic arc whose projective discriminant is ρ=1/2 and its chord is given by

when the control points are (0,0), (a,b) and (d,0).

Approach

Place the conic arc in the position given and use integration to find the area.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Create the conic arc.

Clear[a,b,d];
ca1=ConicArc2D[{0,0},{a,b},{d,0},1/2];

Solve for t in terms of the y-coordinate.

Clear[t];
ans=Solve[ca1[t][[2]]==y,t] //Simplify

Find the x-coordinate of the left side of the rectangle.

X1=ca1[t][[1]] /. ans[[1,1]] //Simplify

Find the x-coordinate of the right side of the rectangle.

X2=ca1[t][[1]] /. ans[[2,1]] //Simplify

Find the width of the rectangle.

L=X2-X1 //Simplify

Find the area by integration (ρ=1/2, so the limits of integration are 0 to b/2).

I1=FullSimplify[Integrate[L,y],Assumptions->{b>0}];
A1=(I1 /. y->b/2)-(I1 /. y->0)