Definite Integral Calculator
Definite integral is a basic concept in calculus that provides a way to calculate the area under a curve at a particular point. This is represented as follows:
Here, a and b are the limits of integration, and f(x) is the function being integrated. The definite integral of f(x) from a to b is the net area between the curve of f(x) and the x-axis within the interval [a, b].
Example 1: Basic Calculation
Consider the function f(x) = x^2. To find the definite integral of f(x) from 1 to 3, we set up the integral as follows:
To solve this, we need to find the antiderivative of x^2, which is (1/3)x^3. Then, we evaluate this antiderivative at the upper and lower limits:
This simplifies to:
Therefore, the definite integral of x^2 from 1 to 3 is 8 2/3.
Example 2: Area Under a Curve
Let's consider another example with the function f(x) = 2x + 1. We want to find the area under the curve from x = 0 to x = 2. The definite integral is set up as:
The antiderivative of 2x + 1 is x^2 + x. Evaluating this at the limits 2 and 0 gives:
Simplifying further, we get:
Thus, the definite integral of 2x + 1 from 0 to 2 is 6.
Properties of Definite Integrals
The definite integral has some important properties:
- Linearity: The integral of a sum is the sum of the integrals. For functions f(x) and g(x), and constants c and d:
- Additivity: If c is a point between a and b, then:
- Reversal of Limits: Reversing the limits of integration changes the sign of the integral:
Understanding how to calculate a particular integral is essential to solving a wide variety of problems in mathematics, physics, engineering, and beyond.