The definite integral gives the **signed**** area** under a curve. Areas above the *x*-axis give a **positive definite** integral. Areas **below** the *x*-axis gives a **negative definite** integral.

We normally think areas are positive, therefore it is important to separate positive and negative integration and use the magnitudes (instead of the signs). We use the absolute value of the definite integral.

Area =

*Remember to sketch the area!!!*

### Examples

1. Find the area enclosed by the curve and the *x*-axis.

So the area is 4.5 square units.

2. Find the area enclosed between the curve , the *x-*axis, and the lines *x*=-1 and *x*=2.

### Show Answer

The sum of areas between *x*=-1 and 0 are calculated separately to *x*=0 and 3.

So the area b = 0.25 square units

So the area a = 20.25 square units.

Hence, the total area = 0.25 + 20.25 = 20.5 square units.

### Area Enclosed by the y-axis

To find the area between a curve and the *y*-axis, we change the subject of the equation to * x*. Therefore:

So the integral is as follows:

### Examples

1. Find the area enclosed by the curve , the *y*-axis and the lines *y*=1 and *y*=3.

Notice that the function *x* is now with variable *y*. Therefore, the curve will become a conic and in this case, a parabolic curve that is rotated 90 degrees to the right (positive).

Therefore, the area is given by:

2. Find the area enclosed between the curve , the *y*-axis, and the lines *y*=0 and *y*=3.

### Show Answer

## Sums and Differences of Areas

In calculus, you could also look for areas between 2 bounded regions. The general method is taking away the primitive values of the upper function (above) by the lower function (below). The resultant value will be the area of the desired region.

### Examples

1. Find the area enclosed between the curve , the *y*-axis, and the lines *y*=0 and *y*=4 in the first quadrant.

Sometimes, you need to find the intersection of the two functions first in order to determine the boundaries.

2. Find the point of intersection of and the line . Hence, find the area enclosed between the two curves.

To find the boundaries we need to use simultaneous equations to solve, then find their primitive functions and insert the values in.