Find Area of Integral Calculator

What is an Integral?

An integral is a mathematical concept that represents the area under a curve. In calculus, integrals are used to find the accumulation of quantities, such as area, volume, and displacement. There are two main types of integrals: definite and indefinite.

Definite integrals calculate the exact area under a curve between two specified points, while indefinite integrals find the antiderivative of a function.

How to Find the Area Under a Curve

To find the area under a curve using calculus, you can use the definite integral. The process involves:

1. Identifying the function and the interval of interest.
2. Setting up the integral with the correct limits of integration.
3. Evaluating the integral to find the exact area.

For functions that are not easily integrable, numerical methods or approximation techniques can be used.

The Integral Formula

This formula calculates the exact area between the curve and the x-axis from \( x = a \) to \( x = b \).

Worked Example

Let's calculate the area under the curve \( f(x) = x^2 \) from \( x = 0 \) to \( x = 2 \).

1. Set up the integral: \( \int_{0}^{2} x^2 \, dx \).
2. Find the antiderivative: \( \frac{x^3}{3} \).
3. Evaluate the antiderivative at the limits:
   • At \( x = 2 \): \( \frac{2^3}{3} = \frac{8}{3} \).
   • At \( x = 0 \): \( \frac{0^3}{3} = 0 \).
4. Subtract the lower limit from the upper limit: \( \frac{8}{3} - 0 = \frac{8}{3} \).

The area under the curve is \( \frac{8}{3} \) square units.