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:

  1. Enter the lower and upper limits of integration.
  2. Specify the number of intervals (must be even).
  3. For more complex functions, enter the function itself.
  4. Click "Calculate" to obtain the Simpson's approximation.
  5. 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.