graphs_and_equations.jpgThe previous topic stated that integration is the opposite of differentiation. Hence its other name of anti-differentiation.

While this is true, one of the main uses of integration is to find the area between a curve and the x-axis. 
These areas are found by evaluating a definite integral.

Defiinite Integrals

A definite integral is calculated by integrating a function between two values. These two values are substituted into the integrated function and the difference taken.

The formula is:

Y11_Area_under_a_Curve_01.gif f(x) dx = F(b) − F(a)

where F(x) is the anti-derivative of f(x)

This complicated looking formula is easier to use than it looks. It simply means doing the following:



Evaluate Y11_Area_under_a_Curve_02.gif(x2 + 3) dx

Step 1 Integrate the function =Y11_Area_under_a_Curve_03.gif
Step 2 Substitute the values 2 and 1

Y11_Area_under_a_Curve_04.gif - Y11_Area_under_a_Curve_05.gif

Step 3 Evaluate


Note that the constant of integration c is not shown as it would be eliminated when subtracting.

Area under a Curve

An accurate way to find the area between a curve, two x-values and the x-axis, as shaded in the example on the diagram below, is to find the definite integral between those x-values.

To find the shaded area between the curve f(x) = x2-6x + 10 , the lines x = 2 and x = 5 and the x-axis, evaluate the definite integral:

Y11_Area_under_a_Curve_07.gif(x2 − 6x + 10) dx



The shaded area is 6 square units.



It is advisable although not essential to draw a diagram before carrying out the integration.

Sometimes, when the curve crosses the x-axis, part of the area may be above the x-axis and part below the x-axis.

When this happens evaluate each area separately, treating both as positive, and then add them together.