When calculating the area under a curve over a certain interval, you can approximate it by getting the area of rectangles under the curve or equal widths. The more narrow the rectangle, the closer the approximation to the area.
The exact area can be found by integrating the function, which is the opposite of differentiation.
Suppose we want the area under the curve defined by the function f(x) = x^2 + 4x + 6 from [1, 4].
We integrate the function first. To do so we take 1/(exponent + 1)(coefficient and variable)^(exponent + 1).
So x^2 becomes (1/3)x^3
4x becomes 2x^2
and 6 becomes 6x
Put it all together we have (1/3)x^3 + 2x^2 + 6x. Now we substitute the values in the interval for x, first 4.
(1/3)(4^3) + 2(4)^2 + 6(4) = 77 1/3
Now substitute 1 for x to get
(1/3) + 4 + 6 = 10 1/3
Subtract 10 1/3 from 77 1/3 to get the final answer of 67.