Math Calculating areas with integrals Area under a curve

# Area under a curve

The area between a curve and the x-axis can be calculated with the definite integral. You should note the following:

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### Remember

For the definite integral, the partitions above the x-axis are positive and those below, negative.

## Definite integral of a positive function

The area under a positive function $f(x)\ge0$ over $[a; b]$ corresponds to the definite integral.

$A=\int_a^b f(x)\,\mathrm{d}x$

## Definite integral of a negative function

The area under a negative function $f(x)\le0$ over $[a; b]$ corresponds to the negative definite integral.

$A=-\int_a^b f(x)\,\mathrm{d}x$

## Definite integral of a function with sign change

The definite integral of a function $f$ with sign change corresponds to the signed area over $[a; b]$.

$A_2-A_1=\int_a^b f(x)\,\mathrm{d}x$

Therefore the area is:

$A=-\int_a^m f(x)\,\mathrm{d}x \,+$ $\int_m^b f(x)\,\mathrm{d}x$