Value of A Function Within An Interval Calculator

What is the Value of a Function Within an Interval?

The value of a function within an interval refers to the definite integral of that function over the specified interval. A definite integral calculates the exact area under the curve of a function between two points, providing a precise value rather than an approximation.

This calculation is fundamental in calculus and has applications in various scientific and engineering fields. For example, in physics, it can represent the total work done by a variable force over a distance, while in economics, it might represent the total consumer surplus.

How to Calculate the Value of a Function Within an Interval

Calculating the value of a function within an interval involves several steps:

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

The result is the exact value of the function within the specified interval.

The Formula

The formula for calculating the value of a function within an interval is:

This formula represents the definite integral of function f(x) from a to b, which equals the difference between the antiderivative evaluated at the upper bound and the antiderivative evaluated at the lower bound.

Worked Example

Let's calculate the value of the function f(x) = x² within the interval [1, 3].

1. Find the antiderivative of f(x): ∫x² dx = (1/3)x³ + C
2. Evaluate at the upper bound (3): (1/3)(3)³ = (1/3)(27) = 9
3. Evaluate at the lower bound (1): (1/3)(1)³ = (1/3)(1) ≈ 0.333
4. Subtract the lower bound value from the upper bound value: 9 - 0.333 ≈ 8.667

The value of the function x² within the interval [1, 3] is approximately 8.667.