Thread: Calculus area

Consider f(x) = x^3 + x^2 + 1 and g(x) = -x^2 + 3x +1
a) Write the integral to compute the area of the region bounded by f and g.

I think you would subtract the bottom function from the top function, but what would the integral of integration be?
Thanks for any help.

First of all you need to solve for where the two curves intersect.

Solve simultaneously:

f(x) = x^3 + x^2 + 1 and g(x) = -x^2 + 3x +1

x^3 + 2x^2 - 3x = 0

x (x^2 + 2x - 3) = 0

x (x + 3)(x - 1) = 0

so the two curves actually intersect at 3 spots... at

x = -3, y = -17
x = 0, y = 1
and x = 1, y = 3

Between -3 and 0, we see that f(x) > g(x) and between 0 and 1, we see that g(x) > f(x)

Therefore, the the area bounded by the two curves should be given by the formula:

integral from -3 to 0 of [f(x) - g(x)] + integral from 0 to 1 of [g(x) - f(x)]

:-)

Hope that helps... and sorry about the notation... the math tags don't seem to be working...