Question

Consider the curves y = x^2and y = mx, where m is some positive constant. No matter what positive constant m is, the two curves enclose a region in the first quadrant.Without using a calculator, find the positive constant m such that the area of the region
bounded by the curves y = x^2 and y = mx is equal to 8.

I would very much appreciate it if someone could find the answer and explain how you did it.

Answer 1

the anti derivative is (m/2)x^2 - 1/3(x)^3 so if you use the 2nd fundamental theorem of calculus using (mx - x^2) over the interval (0,m) and set that equal to 8 (the interval is 0 to m because the function with the larger area is mx) you should get m^3 / 8 = 8. Solve for m and you're done :)