# Calculus kinda with the area of two curves

posted by Roni

II don't even know where to start with this can anyone help?!?
Find c>0 such that the area of the region enclosed by the parabolas
y=x^2-c^2 and y=c^2-x^2 is 270.

1. Steve

the two curves intersect where

x^2-c^2 = c^2-x^2
that is, where x=±c

The area is thus

∫[-c,c] (c^2-x^2)-(x^2-c^2) dx
= 4∫[0,c] c^2-x^2 dx
= 4(c^2 x - 1/3 x^3)[0,c]
= 4(c^3 - 1/3 c^2)
= 8/3 c^3

8/3 c^3 = 270
8c^3 = 30*3^3
c = 3/2 ∛30

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