The Area Between Curve and X Axis Calculator helps users calculate the area enclosed by a curve and the x-axis over a specified interval. This is useful in various fields such as physics, engineering, and economics, where understanding the area under a curve can provide insights into accumulated quantities, total output, and other important measures.

## Formula

To calculate the area between a curve and the x-axis, follow these steps:

- Identify the function f(x) and the interval [a, b] over which you want to find the area.
- Set up the definite integral. The formula for the area A is: A = ∫ from a to b of |f(x)| dx
- Evaluate the integral to find the area.

## Pre-calculated Areas Table

Below is a table with pre-calculated areas for common functions over typical intervals:

Function f(x) | Interval [a, b] | Area A |
---|---|---|

x^2 | [0, 1] | 1/3 |

sin(x) | [0, pi] | 2 |

e^x | [0, 1] | e - 1 |

This table provides a quick reference for common calculations, saving time and effort.

## Example Calculation

Let's calculate the area between the curve f(x) = x^2 and the x-axis over the interval [0, 1]:

- Identify the function f(x) = x^2 and the interval [0, 1].
- Set up the definite integral: A = ∫ from 0 to 1 of x^2 dx
- Evaluate the integral: A = [x^3 / 3] from 0 to 1 = 1/3 - 0/3 = 1/3

Thus, the area between the curve f(x) = x^2 and the x-axis over the interval [0, 1] is 1/3.

## Most Common FAQs

**What is the area under a curve?**

The area under a curve refers to the space between the curve and the x-axis over a specified interval. It can be calculated using definite integrals.

**Why is the absolute value used in the integral?**

The absolute value ensures that the area is always positive, regardless of whether the curve is above or below the x-axis.

**How can this calculator be useful in real life?**

This calculator can be used in various real-life scenarios, such as calculating the total distance traveled by an object.