1. What is Integral Calculation?

The integral of a function represents the area under its curve between two points. In real life, this can represent physical quantities like distance traveled, work done, or total accumulated value.

2. How Does the Calculator Work?

The calculator uses numerical integration methods:

Where:

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

Methods:

• Trapezoidal Rule: Approximates area using trapezoids under the curve
• Simpson's Rule: Uses quadratic approximations for better accuracy

3. Importance of Area Under Curve

Applications: Calculating work in physics, consumer surplus in economics, probability in statistics, and many engineering applications.

4. Using the Calculator

Tips:

• Enter mathematical functions using standard notation (e.g., x^2, sin(x), exp(x))
• Ensure upper limit is greater than lower limit
• For best results, use simple continuous functions

5. Frequently Asked Questions (FAQ)

Q1: What functions can I enter?
A: Basic functions like polynomials (x^2), trigonometric (sin, cos), exponential (exp), and logarithmic (log) functions.

Q2: Why two different methods?
A: Simpson's Rule is generally more accurate but requires more computation. Choose based on your accuracy needs.

Q3: What are common real-world applications?
A: Calculating distance from velocity, work from force, total revenue from marginal revenue, etc.

Q4: How accurate are the results?
A: Accuracy depends on the function and method. Simpson's Rule is typically accurate to 4 decimal places for smooth functions.

Q5: Can I use this for definite integrals?
A: Yes, this calculator computes definite integrals between two points a and b.