Enter the lower limit, upper limit, and number of intervals into the Simpson's approximation calculator to estimate the integral.

## Simpson's Approximation Formula

The Simpson's approximation formula estimates the integral of a function by dividing the interval into multiple sub-intervals and applying a weighted average of function values.

S(x) ≈ h/3 * [f(a) + 4Σ(i=1,n/2-1)(f(a + 2i*h)) + 2Σ(i=1,n/2)(f(a + (2i-1)*h)) + f(b)]

Variables:

- S(x) is the approximate value of the integral
- h is the width of each sub-interval
- f(x) is the function being integrated
- a and b are the lower and upper limits of integration
- n is the number of sub-intervals (must be even)

To use the calculator, input the required values and click "Calculate" to get the Simpson's approximation.

## Understanding Simpson's Approximation

Simpson's approximation is a numerical method used to estimate definite integrals. It provides a more accurate result than methods like the Trapezoidal rule by using parabolic segments instead of straight lines.

## How to Use the Calculator

Follow these steps to calculate an approximation using Simpson's rule:

- Enter the lower and upper limits of integration.
- Specify the number of intervals (must be even).
- For more complex functions, enter the function itself.
- Click "Calculate" to obtain the Simpson's approximation.
- Verify your result for accuracy.

**Example Problem:**

Let's compute the integral using Simpson's approximation with the following values:

Lower Limit (a) = 0

Upper Limit (b) = 2

Number of Intervals (n) = 4

## FAQ

**1. What is Simpson's approximation used for?**

Simpson's approximation is used to estimate definite integrals numerically, providing a more accurate result than simpler methods.

**2. How does Simpson's rule work?**

It approximates the integral of a function by fitting parabolic segments through pairs of points on the curve.

**3. When should Simpson's rule be applied?**

It's beneficial when high accuracy is needed in numerical integration, especially for smooth functions.

**4. Can Simpson's approximation handle complex functions?**

Yes, by entering the function directly into the advanced calculator, Simpson's rule can estimate integrals of various functions.

**5. Is the calculator reliable for all types of integrals?**

The calculator provides a reliable estimate for a wide range of integrals, ensuring accuracy based on the input values.