Help completing this integral.?

Question

This is a problem I saw on answers at some point, but was since removed or some other such thing because my attempts to locate it again have failed despite many attempts with the search option.

The original problem states something as follows:  Three mathematicians order a large 14" pizza (14" being the diameter).  Because they are such great mathematicians, instead of cutting the pizza in the traditional fashion, these mathematicians decide to cut the pizza using only parallel cuts.  What are the locations of these cuts so that each person gets an equal share?

My approach:  Let the pizza be a circle of radius 7 be centered at the origin, equation x^2 + y^2 = 49.  Area of the circle = 49pi ~> each person gets 49/3 pi sq. in. of pizza.
Let the cuts be vertical through the pizza at x = +/- c.  Focus attention on the part of the circle in Quadrant I bound by:
x = 0, x = c, y = 0 and y = sqrt(49 - x^2)
This is 1/4 of the total area given to each person, so it's area is 49/12 pi.

So, the integral I have set up is:

49/12 pi = integral from 0 to c of [sqrt(49 - x^2) dx]

I have not studied how to simplify 49 - x^2 without having an x term already attached to the dx.  Any assistance would be wonderful, or the link the original problem I can no longer locate.

Anonymous · Accepted Answer

The way to do the integral is substitute x = 7 sin t
dx = 7 cos t dt

Integral sqrt(49 - x^2) dx
= int sqrt(49 - 49 sin^2 t) 7 cos t dt
= int 7 cos t . 7 cos t dt
= 49 int cos^2 t dt
= 49 int (cos 2t + 1)/2 dt
= 49 (1/4 sin 2t + t/2)

You can sort out the limits yourself :-)

The bad news is, this leads to an equation of the form

A sin 2t + Bt + C = 0 (where you know the values of A B and C)
There isn't a "simple" way to solve that equation, but you can solve it numerically by graphing the function.

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Help completing this integral.?

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