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 140. C=?

Answer 1

To find the area between two curves you integrate the difference of the top curve and the bottom curve between the points where they intersect each other.

First find where they intersect:

c² - x² = x² - c²

2c² = 2x²

x = c and x = -c

Now we determine which curve is the top curve and which is the bottom:

y = c² - x² is a parabola opening downward and shifted up by c² units.

y = x² - c² is a parabola opening upward and shifted down by c² units.

Drawing this out roughly you'll see that y = c² - x² is above y = x² - c², so we will subtract the lower curve from the higher curve in our integration. Now we can write:

140 = ∫[(c² - x²) - (x² - c²)] dx integrated from -c to c

Simplify:

140 = ∫(2c² - 2x²)dx integrated between -c and c

140 = 2c²x - 2x³/3 from x = -c to x = c

140 = [2c²(c) - 2(c)³/3] - [2c²(-c) - 2(-c)³/3] = [2c³ - 2c³/3] - [-2c³ + 2c³/3] = 8c³/3

Then solve for c:

140 = 8c³/3

c = (105/2)^(1/3) ≈ 3.74