1. What is a Definite Integral?

The definite integral of a function between two limits represents the net area under the curve of that function between those two points. It's a fundamental concept in calculus with applications in physics, engineering, and many other fields.

2. How Does the Calculator Work?

The calculator uses the Fundamental Theorem of Calculus:

Where:

• \( f(x) \) — The function to integrate
• \( F(x) \) — The antiderivative of \( f(x) \)
• \( a \) — Lower limit of integration
• \( b \) — Upper limit of integration

Explanation: The calculator evaluates the antiderivative at both limits and subtracts the lower limit evaluation from the upper limit evaluation.

3. Importance of Definite Integrals

Details: Definite integrals are used to calculate areas, volumes, displacement, work, and many other quantities in mathematics and physics.

4. Using the Calculator

Tips: Enter the antiderivative formula (like "x^2/2" for integrating x), then enter the lower and upper limits. The calculator will compute the definite integral.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between definite and indefinite integrals?
A: A definite integral has limits and gives a numerical value, while an indefinite integral gives a family of functions (the antiderivatives).

Q2: Can this calculator handle any function?
A: This simplified version handles basic polynomials. A full implementation would need a sophisticated math parser.

Q3: What if my function doesn't have an elementary antiderivative?
A: Some functions require numerical integration methods (like Simpson's rule) rather than symbolic integration.

Q4: What does the result represent geometrically?
A: It represents the net area between the curve and the x-axis from a to b (accounting for areas below the axis as negative).

Q5: Can definite integrals be negative?
A: Yes, when the curve is below the x-axis between the limits, the integral will be negative.