Belirli Integral Calculator

What is a Definite Integral?

The definite integral of a function f(x) with respect to x, from a to b, is written as ∫[a,b] f(x) dx. It calculates the exact area under the curve of f(x) between x = a and x = b.

Definite integrals have many applications in physics, engineering, economics, and other sciences where accumulation of quantities is important.

How to Calculate a Definite Integral

To calculate a definite integral:

1. Identify the function f(x) you want to integrate
2. Determine the lower limit (a) and upper limit (b)
3. Find the antiderivative F(x) of f(x)
4. Evaluate F(x) at the upper limit and lower limit
5. Subtract the lower limit evaluation from the upper limit evaluation

The result is the exact area under the curve between the two limits.

Formula

The antiderivative F(x) is found by reversing the differentiation process. Common antiderivatives include:

• ∫x^n dx = (x^(n+1))/(n+1) + C (for n ≠ -1)
• ∫sin(x) dx = -cos(x) + C
• ∫cos(x) dx = sin(x) + C
• ∫e^x dx = e^x + C

Example Calculation

Let's calculate the definite integral of f(x) = x² from 0 to 2.

1. Find the antiderivative: ∫x² dx = (x³)/3 + C
2. Evaluate at upper limit: (2³)/3 = 8/3
3. Evaluate at lower limit: (0³)/3 = 0
4. Subtract: (8/3) - 0 = 8/3 ≈ 2.6667

The area under the curve x² from 0 to 2 is 8/3 square units.