Math Application integral calculus Calculating areas with integrals

# Area under a curve

The area between a curve and the x-axis can be calculated with the definite integral.

Here is a short overview. You can find more information about calculating areas with integrals in the corresponding article.

!

### Remember

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

## Functions without change of sign

The area under a function $f(x)$ without change of sign over $[a; b]$ corresponds to the absolute value of the definite integral.

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

## Functions with change of sign

The area is partly above and partly below the x-axis. The divided areas must therefore be calculated and added separately.

$A=A_1+A_2$

## Area between curves

The enclosed area in the interval $[a; b]$ is calculated by the formula:

$A=|\int_a^b (f(x)-g(x))\,\mathrm{d}x|$
The text passage provides an overview of how to calculate areas using definite integrals, including areas under a curve, areas for functions with and without a change of sign, and areas between curves. It does not contain any specific mathematical questions or answers to such questions.
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Math Application integral calculus Calculating areas with integrals

# Area under a curve

The area between a curve and the x-axis can be calculated with the definite integral.

Here is a short overview. You can find more information about calculating areas with integrals in the corresponding article.

!

### Remember

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

## Functions without change of sign

The area under a function $f(x)$ without change of sign over $[a; b]$ corresponds to the absolute value of the definite integral.

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

## Functions with change of sign

The area is partly above and partly below the x-axis. The divided areas must therefore be calculated and added separately.

$A=A_1+A_2$

## Area between curves

The enclosed area in the interval $[a; b]$ is calculated by the formula:

$A=|\int_a^b (f(x)-g(x))\,\mathrm{d}x|$
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The text passage provides an overview of how to calculate areas using definite integrals, including areas under a curve, areas for functions with and without a change of sign, and areas between curves. It does not contain any specific mathematical questions or answers to such questions.

TYPE: NO_QUESTION