## Area under Curves

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 gives the area shaded on the right. ### 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.

 Example 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: (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.

 Example Find the shaded area: Evaluate:  ### 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

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