Composite Trapezoidal Rule Calculator Integral

Composite Trapezoidal Rule:

\[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{2} \left[ f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right] \]

Approximate Integral:

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1. What is the Composite Trapezoidal Rule?

The Composite Trapezoidal Rule is a numerical integration technique that approximates the definite integral of a function by dividing the area under the curve into multiple trapezoids and summing their areas. It provides a more accurate approximation than the simple trapezoidal rule by using multiple intervals.

2. How Does the Calculator Work?

The calculator uses the Composite Trapezoidal Rule formula:

\[ \int_{a}^{b} f(x) \, dx \approx \frac{h}{2} \left[ f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right] \]

Where:

\( a \) — Lower limit of integration
\( b \) — Upper limit of integration
\( n \) — Number of intervals
\( h = \frac{b - a}{n} \) — Width of each interval
\( x_i = a + i \times h \) — Points where the function is evaluated

Explanation: The method approximates the area under the curve by summing the areas of n trapezoids, providing better accuracy as n increases.

3. Importance of Numerical Integration

Details: Numerical integration is essential when functions cannot be integrated analytically or when dealing with empirical data. The Composite Trapezoidal Rule provides a simple yet effective method for approximating definite integrals with known error bounds.

4. Using the Calculator

Tips: Enter the function using standard mathematical notation (e.g., x^2, sin(x), exp(x)), specify the integration limits, and choose an appropriate number of intervals. More intervals generally yield more accurate results but require more computation.

5. Frequently Asked Questions (FAQ)

Q1: How accurate is the Composite Trapezoidal Rule?
A: The error is proportional to (b-a)³/n², meaning doubling the number of intervals reduces the error by a factor of 4.

Q2: When should I use this method?
A: It's suitable for functions that are relatively smooth over the integration interval. For functions with rapid changes, other methods like Simpson's rule may be more appropriate.

Q3: What functions can I input?
A: You can use basic mathematical operations (+, -, *, /, ^) and common functions like sin, cos, tan, exp, log, sqrt. Use 'x' as the variable.

Q4: How many intervals should I use?
A: Start with 10-20 intervals and increase if more precision is needed. More intervals give better accuracy but require more computation.

Q5: Can this method handle improper integrals?
A: The basic Composite Trapezoidal Rule is not suitable for improper integrals. Special techniques are needed for integrals with infinite limits or singularities.