Calculating Definite Integral

A definite integral calculates the exact area under a curve between two specified points. This guide explains how to calculate definite integrals, provides an interactive calculator, and covers common functions and applications.

What is a Definite Integral?

A definite integral represents the signed area between a function's curve and the x-axis over a specified interval [a, b]. It provides exact values for quantities like area, distance, volume, and work.

The definite integral of a function f(x) from a to b is written as:

∫_a^b f(x) dx

This notation indicates the accumulation of f(x) from x = a to x = b.

How to Calculate a Definite Integral

Step 1: Find the Antiderivative

First, find the antiderivative (indefinite integral) of the function f(x). This is a new function F(x) such that F'(x) = f(x).

Step 2: Apply the Fundamental Theorem of Calculus

Use the antiderivative to evaluate the definite integral:

∫_a^b f(x) dx = F(b) - F(a)

This calculates the net area between the curve and the x-axis from a to b.

Example Calculation

Calculate ∫₀² 3x² dx:

Find the antiderivative: ∫3x² dx = x³ + C
Apply the limits: (2³) - (0³) = 8 - 0 = 8

The definite integral is 8.

Common Functions and Their Integrals

Function	Antiderivative	Example
xⁿ	(xⁿ⁺¹)/(n+1) + C (n ≠ -1)	∫x² dx = x³/3 + C
eˣ	eˣ + C	∫eˣ dx = eˣ + C
sin(x)	-cos(x) + C	∫sin(x) dx = -cos(x) + C
cos(x)	sin(x) + C	∫cos(x) dx = sin(x) + C

Applications of Definite Integrals

Calculating areas between curves
Finding volumes of revolution
Determining work done by variable forces
Computing average values of functions
Analyzing population growth rates

Frequently Asked Questions

What's the difference between definite and indefinite integrals?

A definite integral calculates the exact area under a curve between two points, while an indefinite integral finds the general antiderivative with an arbitrary constant.

How do I know when to use definite integrals?

Use definite integrals when you need exact values for quantities like area, distance, or work over a specific interval.

Can definite integrals be negative?

Yes, definite integrals can be negative when the function is negative over the interval, representing a net area below the x-axis.