6FAreaUnderCurves


 * Area Under Curves**

Area above x-axis To find the area between the **x-axis**, the lines **x = a**, **x = b** and the curve **//y = f(x)//** .... Use the definite integral

math . \quad \text{Area } \; = \displaystyle{\int\limits_{x=a}^{x=b} f(x) }\; dx math

Area below x-axis ​ .... The definite integral math . \quad \displaystyle{\int\limits_{x=a}^{x=b} f(x) } \; dx math .... gives a __negative__ result.

math .\quad \text{Area } \; = \left| \displaystyle{\int\limits_{x=a}^{x=b} f(x) } \;dx \right| math


 * Note: ** The definite integral is negative but the __**area**__ is positive.

Area above and below x-axis

Note that the area from b to c will be negative so the area must be calculated in two sections.

math . \quad \text{Area } \; = \displaystyle{\int\limits_{x=a}^{x=b} f(x) }\; dx + \left| \displaystyle{\int\limits_{x=b}^{x=c} f(x) } \; dx \right| math

Example

Area between y-axis and curve

The area between the **y-axis**, the lines **y = a**, **y = b** and **//y = f(x)//** {rearrange y = f(x) to make x the subject}

math . \quad \text{Area } \; = \displaystyle{\int\limits_{y=a}^{y=b} x } \; dy math

Example

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