Definite Integrals Calculator

What is a Definite Integral?

A definite integral calculates the exact area under a curve between two specified points, denoted by the limits of integration. Unlike indefinite integrals, which represent a family of functions, definite integrals provide a single numerical value.

The concept of definite integrals was first formalized by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. They recognized that the area under a curve could be calculated by summing infinitesimally small rectangles, leading to the fundamental theorem of calculus.

How to Calculate a Definite Integral

Calculating a definite integral involves several steps:

1. Identify the function to be integrated and the limits of integration (lower and upper bounds).
2. Find the antiderivative (indefinite integral) of the function.
3. Evaluate the antiderivative at the upper limit and subtract its value at the lower limit.

For example, to find the area under the curve of f(x) = x² from x = 0 to x = 2:

1. Find the antiderivative of x², which is (x³)/3.
2. Evaluate at the upper limit: (2³)/3 = 8/3.
3. Evaluate at the lower limit: (0³)/3 = 0.
4. Subtract the lower value from the upper value: 8/3 - 0 = 8/3.

The Definite Integral Formula

This formula is derived from the fundamental theorem of calculus, which connects differentiation and integration. The antiderivative F(x) must be continuous on the closed interval [a, b].

Worked Examples

Example 1: Linear Function

Calculate ∫[1 to 3] (2x + 1) dx.

1. Find the antiderivative: ∫(2x + 1) dx = x² + x + C.
2. Evaluate at upper limit: (3)² + 3 = 9 + 3 = 12.
3. Evaluate at lower limit: (1)² + 1 = 1 + 1 = 2.
4. Subtract: 12 - 2 = 10.

The definite integral is 10.

Example 2: Exponential Function

Calculate ∫[0 to ln(2)] e^x dx.

1. Find the antiderivative: ∫e^x dx = e^x + C.
2. Evaluate at upper limit: e^(ln(2)) = 2.
3. Evaluate at lower limit: e^0 = 1.
4. Subtract: 2 - 1 = 1.

The definite integral is 1.

Applications of Definite Integrals

Definite integrals have numerous practical applications:

• Calculating areas under curves in physics and engineering.
• Determining the work done by a variable force in physics.
• Finding the average value of a function over an interval.
• Computing probabilities in statistics.
• Modeling population growth in biology.

In physics, definite integrals are used to calculate the center of mass of a variable-density object. In finance, they help determine the present value of a future payment stream.