Find the are of the region bound by the graphs..?

Question

Find the are of the region bound by the graphs of:
f(x) = 2bx - x^2 AND g(x) = bx

Answer 1

Areas of regions are integration problems. All that's missing is the boundaries for these two curves. That's not too hard to find though - set them equal to each other and solve for x:

2bx - x^2 = bx
bx - x^2 = 0
x(b - x) = 0

x = 0

b - x = 0
x = b

Now that we have boundaries, let's integrate (I'm using S as the integration symbol):

A = S [f(x) - g(x)] dx
A = S [(2bx - x^2) - bx] dx
A = S (bx - x^2) dx
A = (bx^2)/2 - (x^3)/3, evaluated from 0 to b
A = [(b(b^2))/2 - (b^3)/3] - [(b(0^2))/2 - (0^3)/3]
A = (b^3)/2 - 9b^3)/3
A = (b^3)/6