Trapezoidal Rule Calculator

Trapezoidal Rule:

\[ \int_{a}^{b} f(x) dx \approx \frac{h}{2} \left( f_0 + 2 \sum_{i=1}^{n-1} f_i + f_n \right) \]

Approximate Integral:

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1. What is the Trapezoidal Rule?

The Trapezoidal Rule is a numerical integration technique that approximates the definite integral of a function by dividing the area under the curve into trapezoids. It provides a simple yet effective method for estimating integrals when an analytical solution is difficult or impossible.

2. How Does the Calculator Work?

The calculator uses the Trapezoidal Rule formula:

\[ \int_{a}^{b} f(x) dx \approx \frac{h}{2} \left( f_0 + 2 \sum_{i=1}^{n-1} f_i + f_n \right) \]

Where:

\( f(x) \) — The function to integrate
\( a \) — Lower limit of integration
\( b \) — Upper limit of integration
\( n \) — Number of intervals
\( h = \frac{b-a}{n} \) — Step size
\( f_i = f(a + i \cdot h) \) — Function value at each point

Explanation: The method approximates the area under the curve by summing the areas of trapezoids formed between each pair of points.

3. Importance of Numerical Integration

Details: Numerical integration is essential when dealing with functions that don't have elementary antiderivatives, or when working with empirical data points rather than functional forms.

4. Using the Calculator

Tips:

Enter the function using PHP mathematical syntax (e.g., "x^2 + 3*x - 5")
Ensure the upper limit is greater than the lower limit
Use more intervals for better accuracy (but more computation time)

5. Frequently Asked Questions (FAQ)

Q1: How accurate is the Trapezoidal Rule?
A: The error is proportional to \( (b-a)^3/n^2 \). Doubling n reduces error by about a factor of 4.

Q2: When should I use this instead of other methods?
A: The Trapezoidal Rule is simple and works well for smooth functions. For oscillatory functions, other methods like Simpson's Rule may be better.

Q3: What are common pitfalls?
A: Using too few intervals for rapidly changing functions, or not properly handling singularities.

Q4: Can I use this for improper integrals?
A: Not directly. Special techniques are needed for integrals with infinite limits or singularities.

Q5: How does this compare to Riemann sums?
A: The Trapezoidal Rule typically converges faster than left/right Riemann sums and is often more accurate for the same number of intervals.