09Fsignedarea


 * Negative Area **



If the region being investigated is __**below**__ the x-axis, then the definite integral will give a __**negative**__ answer.


 * {Because f(x) is negative, f(x) × h will be negative} **

But __**Area**__ should always be __**positive**__!

Hence, if the curve is below the x-axis:


 * Example 1 **

Find (to 3 decimal places) the area between the x-axis and the curve y = x 2 – 3 between the values x = 0.5 and x = 1.5

__Solution__
 * {A sketch graph reveals curve is below axis in the domain [0.5, 1.5] **

Definite integral is: math \\ . \qquad \displaystyle{ \int\limits_{0.5}^{1.5} \,x^2-3 \; dx} \qquad. \\ . \\ . \qquad = \left[ \frac{1}{3}x^3 - 3x \right]_{0.5}^{1.5} \qquad. \\ .\\ . \qquad = \big( -3.375 \big) - \big( -1.458 \big) \qquad. \\ . \\ . \qquad = -1.917 math

Hence Area = +1.917 square units.

Reversing Terminals

Reversing the order of the terminals (or limits) reverses the sign of the definite integral

This result can sometimes be useful when manipulating areas.

Combining Regions

If the curve crosses the x-axis within the domain, the definite integral over the entire domain will __**not**__ give the correct __**area**__. This is because the negative section and the positive section will cancel each other out.

The solution is to calculate each section seperately and add the results:

Find (correct to 3 decimal places) the area between the function y = 2sin(x – 1.5) and the x-axis and between x = 0 and x = 2. NOTE: math . \qquad \displaystyle{ \int\limits_0^2 2\sin(x-1.5)\, dx} = -1.614 \qquad \textit{Incorrect Area!!} \qquad. math
 * Example 2 **

__Solution__
 * {A sketch graph reveals curve has an x-intercept at x = 1.5} **

math \\ . \qquad \textbf{Area } = \left| \int\limits_0^{1.5} 2\sin(x-1.5)\, dx \right| + \left| \int\limits_{1.5}^2 2\sin(x-1.5)\, dx \right| \qquad. \\ . \\ . \qquad \textbf{Area } = \left| -1.859 \right| + \left| 0.245 \right| \\. \\ . \qquad \textbf{Area } = 1.859 + 0.245 \\ .\\ . \qquad \textbf{Area } = 2.104 \; \textit{ square units} \qquad. math .