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 72.

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embizze | High School Teacher | (Level 1) Educator Emeritus

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Find `c>0` such that the area between the curves `y=x^2-c^2` and `y=c^2-x^2` is 72.

(1) The graphs:

Note that the intersections of the graphs are at (-c,0) and (c,0). Also note the symmetry about the x-axis. Thus we need only find the area bounded by the x-axis and the curve `y=c^2-x^2` to be 36.

The limits of integration will be -c,c.

(2) `int_(-c)^c c^2-x^2=c^2x-x^3/3 |_(-c)^c`

`=(c^3-(-c^3))-(c^3/3-(-c^3/3))`

`=4/3 c^3`

This represents the area bounded by the x-axis and the curve `y=c^2-x^2` from -c to c. The area of the region of interest is twice this area.

(3) `2(4/3)c^3=72`

`8/3 c^3=72`

`c^3=27`

`c=3`

Therefore, for c=3 we have the area between `y=x^9-9` and `y=9-x^2` is 72 as required.

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