Bounded Integral Calculator

What is a Bounded Integral?

A bounded integral, also known as a definite integral, calculates the area under a curve between two specified points (the bounds). Unlike indefinite integrals, which represent a family of functions, definite integrals yield a single numerical value.

Bounded integrals are fundamental in calculus for solving problems involving accumulation, such as finding areas, volumes, and work done by a variable force.

How to Calculate a Bounded Integral

To compute a bounded integral, follow these steps:

1. Identify the function to integrate and the upper and lower bounds.
2. Find the antiderivative (indefinite integral) of the function.
3. Evaluate the antiderivative at the upper bound and subtract its value at the lower bound.

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

The Bounded Integral Formula

This formula calculates the exact area under the curve of f(x) between x = a and x = b.

Worked Example

Let's calculate the bounded integral of f(x) = x² from x = 1 to x = 3.

1. Find the antiderivative of x²: ∫x² dx = (1/3)x³ + C
2. Evaluate at the upper bound (x = 3): (1/3)(3)³ = 9
3. Evaluate at the lower bound (x = 1): (1/3)(1)³ = 1/3
4. Subtract the lower evaluation from the upper: 9 - (1/3) = 26/3 ≈ 8.6667

The area under the curve x² from x = 1 to x = 3 is 26/3 square units.

Applications of Bounded Integrals

Bounded integrals have numerous practical applications in various fields:

• Calculating areas between curves in physics and engineering
• Determining volumes of solids in geometry
• Computing work done by variable forces in mechanics
• Finding average values in statistics
• Modeling population growth in biology