Consider a graph of (y=f(x)). The area under the curve to the left of (x) can be defined as a function of (x) called (A(x)).

mtaimg/images/topics/8/8-93-1.png/mtaimg

If (x) increases by a small amount (\delta x), the area increases by an amount (\delta A ).

This increase (\delta A) is approximately rectangular and is approximately (y \delta x).

(\delta A\ ≈ y \delta x\ \implies \dfrac{\delta A}{\delta x} ≈ y )

As (\delta x→0), any error between (y \delta x) and the actual area will be negligible.

(\lim\limits_{\delta x→0} \dfrac{\delta A}{\delta x} = y \implies \dfrac{dA}{dx} = y )

Integrating both sides with respect to (x) gives:

(\displaystyle\int{\dfrac{dA}{dx}} dx = \displaystyle\int{y} dx \implies A = \displaystyle\int{y} dx)

Consider the graph below. The area bounded by (y=f(x)), (x=a), (x=b) and the (x)-axis can be approximated by calculating the sum of many rectangles.

mtaimg/images/topics/8/8-93-2.png/mtaimg

The sum of all (n) rectangles can be written as:

(Area ≈ f(x_1)\delta x + f(x_2)\delta x + ... + f(x_n)\delta x )

(\implies Area ≈ \displaystyle\sum_{^n{f(x_k)\delta} x} )

This value can be made more accurate by using a large number of very thin rectangles. The exact area can be found when (n→∞) and (\delta x → 0).

(Area = \lim\limits_n→∞ \displaystyle\sum_^n f(x_k) \delta x )

The lower and upper limits on the sum correspond to the first and last rectangle where (x=a) and (x=b) respectively, so this could also be written as:

(Area = \lim\limits_{\delta x→0} \displaystyle\sum_^b f(x) \delta x )

This is equivalent to notation for definite integration:

(\boxed{\displaystyle\int_a^b{f(x)} dx = \lim\limits_{\delta x→0} \displaystyle\sum_^b f(x) \delta x} )

[b]uIntegration as the limit of a sum[/u]/b

(\displaystyle\int_a^b{f(x)} dx = \lim\limits_{\delta x→0} \displaystyle\sum_^b f(x) \delta x )