Area Under a Curve:

Properties of Areas:

1. Given any region R, then the area, A(R) is a real number;
   A(R) ³ 0.
2. Given two congruent regions, then their areas are equal.
3. If R = R1 È R2 ,where R1 and R2 have only boundary points in common, then A(R1) + A(R2) = A(R).
4. To assign a real number to an area of a region is to consider a very simple area of a region such as the area of a rectangle; A = l·w.

With the four given properties above; the area of any shape can be approximated by using the limiting process.

The area of the curve bounded by the interval [a,b], the x-axis and the graph of the function.

1. Divide the interval into n equal subintervals, each subinterval has a length of

2. Find the ith subinterval of the interval.

x₀ = a
x₁ = a + Dx
x₂ = a + 2·Dx
x₃ = a + 3·Dx
x_i = a + i·Dx
x_n = b

3. After the ith interval has been found, the area can be approximated from the limiting process, the area of a rectangle and the summation process. The formula is given as:

where Dx is the base and f(xi) is the height of the rectangle.

The area then can be calculated by the use of the properties of sums and the sum of n powers, then taking the limit as n goes to infinity.