Evaluate Each Definite Integral Calculator

Definite integrals calculate the exact area under a curve between two points. This calculator evaluates definite integrals accurately and explains the process step-by-step.

What is a Definite Integral?

A definite integral calculates the exact area under a curve between two specified limits. It's represented as:

∫_a^b f(x) dx = F(b) - F(a)

Where:

f(x) is the integrand (the function to integrate)
a and b are the lower and upper limits of integration
F(x) is the antiderivative of f(x)

Definite integrals have practical applications in physics, engineering, economics, and many other fields.

How to Evaluate a Definite Integral

Follow these steps to evaluate a definite integral:

Identify the integrand (f(x)) and the limits of integration (a and b)
Find the antiderivative F(x) of f(x)
Evaluate F(x) at the upper limit (F(b))
Evaluate F(x) at the lower limit (F(a))
Subtract the two results: F(b) - F(a)

Note: The antiderivative F(x) must be continuous on the interval [a, b].

Common Integral Formulas

Here are some fundamental integral formulas you should know:

Integrand	Antiderivative
xⁿ (n ≠ -1)	(xⁿ⁺¹)/(n+1) + C
1/x	ln\|x\| + C
e^x	e^x + C
sin(x)	-cos(x) + C
cos(x)	sin(x) + C

Example Calculations

Let's evaluate the integral of x² from 0 to 2:

∫₀² x² dx

Step 1: Find the antiderivative of x²

∫x² dx = (x³)/3 + C

Step 2: Evaluate at the upper limit (x=2)

(2³)/3 = 8/3

Step 3: Evaluate at the lower limit (x=0)

(0³)/3 = 0

Step 4: Subtract the two results

(8/3) - 0 = 8/3 ≈ 2.6667

The area under the curve x² from 0 to 2 is approximately 2.6667.

Interpreting Results

The result of a definite integral represents:

The exact area under the curve between the specified limits
The net accumulation of the quantity represented by the integrand
The total change in the function over the interval

For example, in physics, the integral of velocity over time gives displacement. In economics, it calculates total revenue or cost.

FAQ

What's the difference between definite and indefinite integrals?: A definite integral has specific limits of integration and calculates an exact area or quantity. An indefinite integral finds the antiderivative without limits.
How do I know if I've found the correct antiderivative?: Differentiate your antiderivative to check if you get back to the original function. If you're unsure, use our integral calculator to verify.
Can definite integrals be negative?: Yes, if the function is negative over part of the interval, the integral can be negative. The result represents the net area.
What if my function doesn't have a known antiderivative?: For complex functions, numerical methods or approximation techniques may be needed. Our calculator handles basic functions well.
How precise are the results from this calculator?: The calculator provides results with up to 6 decimal places for accuracy. For most practical purposes, this is sufficient.