Find the value(s) of c such that the area of the region bounded by the parabolae y=x^2−c^2 and y = c^ 2 − x ^2 is 1944

Answer 1

You have two intersecting parabolas and you want to find the area by integrating from the lower point of intersection to the higher point. The function integrated will be the f(x) = y_hi - y_lo

Points of intersection of the two curves are such that x² - c² = c² - x² , so x = +/- c

The area will be the integral from -c to c of (c² - x² ) - (x² - c²) or 2(c² - x²). You could use symmetry to integrate from 0 to c and just (c² - x²), but you would then have to remember to divide the area by 4.

The integral yields 2(c²x - x³/3) evaluated at c and -c. This equals the area and you can solve for c.

Good luck!

Answer 2

The first step is to figure out the x values where these two curves intersect so we can use those as the bounds for our integral. You can do this by setting them equal to each other.

x² - c² = c² - x²

2x² = 2c²

x² = c²

x = ± c

So the area between the curves is given by the integral of the higher function minus the lower function with bounds -c and c.

1944 = ∫_-c^c c² - x² - (x² - c²) dx

So you can simplify and evaluate that integral with respect to x and you should be left with an equation containing only c. Then you can solve for c.