Use the integral calculator to find the area under a curve defined by a function. Enter the function and limits to calculate the definite integral.

Integral Calculation Formula

The following formula is used to compute the definite integral of a function f(x) over the interval [a, b]:

∫ab f(x) dx = F(b) - F(a)

Where:

  • ∫ab f(x) dx represents the definite integral of the function f(x) from a to b.
  • F(x) is the antiderivative of f(x), meaning F'(x) = f(x).
  • F(b) and F(a) are the values of the antiderivative evaluated at b and a, respectively.

To calculate the definite integral, find the antiderivative of f(x), evaluate it at the upper limit, and subtract its value evaluated at the lower limit.

What is an Integral?

An integral in calculus represents the area under a curve or the accumulation of quantities. There are two main types of integrals: definite integrals, which calculate the area under a curve between two points, and indefinite integrals, which find the general form of the antiderivative of a function.

How to Calculate an Integral?

Follow these steps to calculate a definite integral:

  1. Identify the function f(x) you want to integrate.
  2. Determine the limits of integration (a and b).
  3. Find the antiderivative F(x) of the function f(x).
  4. Evaluate the antiderivative at the upper limit (b) and the lower limit (a).
  5. Subtract the value at the lower limit from the value at the upper limit to get the result.
  6. Enter your function and limits into the calculator above to verify your results.

Example Problem:

Calculate the integral of the function f(x) = 2x from 1 to 3.

Function: f(x) = 2x

Lower Limit: a = 1

Upper Limit: b = 3

FAQ

1. What is an antiderivative?

An antiderivative of a function f(x) is another function F(x) such that F'(x) = f(x). It is also known as an indefinite integral.

2. How is a definite integral different from an indefinite integral?

A definite integral calculates the area under a curve within specific limits, while an indefinite integral represents a family of functions without limits.

3. What methods can be used for advanced integration?

Advanced integration techniques include substitution, integration by parts, and numerical methods such as Simpson’s rule or trapezoidal rule.

4. Can integrals be calculated without a calculator?

Yes, integrals can be solved analytically using calculus techniques, but for complex functions or definite integrals, using a calculator or software is often more practical.

5. Is the integral calculator accurate?

The integral calculator provides an estimate based on the input function and limits. For exact results, especially with complex functions, consult advanced mathematical tools or software.