Calculate Definite Integral Using The Definition

A definite integral calculates the exact area under a curve between two points. This guide explains how to compute it using the definition of integration as a limit of Riemann sums.

What is a Definite Integral?

A definite integral represents the signed area between a function's graph and the x-axis over a specified interval [a, b]. It provides exact values for quantities like total distance traveled, accumulated work, or total change in a quantity.

Mathematically, the definite integral of f(x) from a to b is written as:

Definite Integral Formula

∫[a,b] f(x) dx = lim(n→∞) Σ[f(x_i*)Δx], where Δx = (b-a)/n

This formula represents the limit as the number of subintervals (n) approaches infinity of the sum of function values multiplied by subinterval widths.

How to Calculate a Definite Integral Using the Definition

Calculating a definite integral using the definition involves these steps:

Divide the interval [a, b] into n equal subintervals of width Δx = (b-a)/n
Choose sample points x_i* in each subinterval (left, right, or midpoint)
Calculate f(x_i*) for each sample point
Sum the products f(x_i*)Δx for all subintervals
Take the limit as n approaches infinity

Practical Approach

In practice, we approximate by using a large number of subintervals (e.g., n = 1000 or 10,000) rather than taking the exact limit.

Example Calculation

Let's calculate ∫[1,3] x² dx using the definition with n = 10 subintervals and right endpoints.

Δx = (3-1)/10 = 0.2
Sample points: 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 2.8, 3.0
Calculate f(x_i*) = x² for each point
Sum: (1.2² + 1.4² + ... + 3.0²) × 0.2 ≈ 10.84

The exact value is 9, showing how increasing n improves accuracy.

Interpreting the Result

The result represents the net area under the curve between the specified limits. For positive functions, this is the total area. For functions that cross the x-axis, the result may be negative if the area below the x-axis dominates.

Applications include:

Calculating total distance traveled by an object with varying speed
Determining total work done by a variable force
Finding accumulated change in a quantity over time

Frequently Asked Questions

What's the difference between definite and indefinite integrals?

A definite integral calculates a specific area between limits, while an indefinite integral finds the antiderivative (family of functions) that represents the area up to any point.

How does increasing n affect the accuracy?

Increasing the number of subintervals (n) makes the Riemann sum approximation more accurate, approaching the exact integral value as n approaches infinity.

When would I use this method versus antiderivative rules?

This method is useful when the antiderivative is difficult to find or when you need to understand the integration process. For most functions, antiderivative rules are more efficient.