Find the exact positive value of c if the area between the graph of y=x^2−c^2 and the x-axis is 42.

Answer 1

To find integration limits, you need to find intersection points of given function and x-axis, and then calculate area = definite integral of f(x) with those integration limits; the next step, use equality that integral and given area (42 units), in order to find "c".

Answer 2

Integration limits x² - c² = 0; x = ± c;

A = ∫_-c^c(c² - x²)dx = 2∫₀^c(c² - x²)dx = 2(c²x - x³/3)₀^c = 4c³/3 = 42; c³ = 63/2; c = (63/2)^1/3