We can use the anti-derivative or integral to find the area under a curve between two points.  For example, suppose we want to find the area under the curve defined by the function f(x) = x^3 + 2x^2 - 4x + 6 between x = 1 and x = 4.

First we integrate the function. 

(1/4)x^4 + (2/3)x^3 - 2x^2 + 6x..

Now we evaluate the integral at 4 and then at 1 and subtract.

(1/4)(4)^4 + (2/3)(4)^3 - 2(4)^2 + 6(4) = 98.6666666666

(1/4)(1)^4 + (2/3)(1)^3 - 2(1)^2 + 6(1) = 4.91666666666

Subtract the two to get the area of 93.75