Differentiation gives the gradient of a curve at a particular point.

Integration has a wide range of applications incuding finding volumes, heat, mass and areas.

In earlier topics, we estimated the area under curves using the Trapezium Rule and Simpson's Rule, we now study a more precise method.

The definite integral Y12_Area_under_Curves_01.gif gives the area shaded on the right. Y12_Area_under_Curves_02.gif

Area between Curve and x-axis

To find the area between a curve, two x-values and the x-axis, as shaded in the example on the diagram below, evaluate the definite integral between those x-values.


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:

Y12_Area_under_Curves_03.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.

Area below x-axis

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.

The absolute (positive) signs indicate this in the example below.


Find the shaded area:




Area between Two Curves

If the area between two curves is required then the following rule works well.

Area between two functions =  (top function − lower function) dx


Find the shaded area between the functions 
y = x2 and y = 2x − x2.