Simpson's Rule Examples

Simpson’s rule calculator 

Simpson’s rule calculator provides the complete procedure to find the closest value to the area of a definite integral

This particular tool is known as Simpson’s ⅓ rule calculator because it uses second-degree polynomials. 

How to use this calculator?

Operating the Simpson's Rule calculator is a seamless process, involving these steps:

  1. Input the function in the provided field.
  2. Specify the values for 'a' and 'b,' which define the interval's starting and endpoints.
  3. Input the desired number of sub-intervals, 'n.'
  4. Click 'Calculate' to receive the result in no time.

What is Simpson’s rule?

Simpson's Rule is a numerical integration method used to estimate the definite integral of a function over a specified interval. It achieves this by approximating the integral using a series of quadratic polynomials or parabolas that best fit the given function.

Simpson's Rule is favored over other methods like Riemann's Sum or the Midpoint Rule because it provides more accurate approximations. 

This increased accuracy stems from its use of parabolic segments rather than rectangles to approximate the area beneath the curve. 

Why do we need it?

Not all functions can be integrated using antiderivative techniques. Some functions like scientific experiments with observed readings require a different method. This is where Simpson’s rule is used. 

Simpson's Rule is particularly suitable for estimating the area under smooth, continuous functions and functions with a moderate degree of curvature.

Simpson’s rule formula:

Simpson's Rule can be summarized as follows:

∫(a, b) f(x) dx ≈ (h/3) [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + ... + 2f(xn-2) + 4f(xn-1) + f(xn)]


Simpson's Rule is derived from the quadratic approximation concept. It requires the following steps:

1-4-1 Approximation:  A graph is divided into equal intervals with endpoints and a midpoint of half-width 'h.'

This division permits the calculation of the function values i.e y-axis values (yL, yC, and yR), which are essential for the formula. Using these values, the area under the quadratic curve intersecting these points is represented as:

= h/3 (yL + 4yC + yR)

This value closely approximates the area beneath the curve.

Implementation: Divide the function into many parabolic curves. Each curve should provide a reasonable approximation of the parabolic area under it. 

The following steps detail the derivation of Simpson's Rule to estimate the area under irregular graphs:

  1. Divide the interval [a, b] into an even number (n) of equally spaced sub-intervals, resulting in n+1 (even) equally spaced points (x0, x1, x2, ..., xn), with a constant width of h = (b-a)/n.
  2. For each pair of adjacent sub-intervals, construct a quadratic function (parabola) that passes through the three points (x0, f(x0)), (x1, f(x1)), and (x2, f(x2)).
  3. Calculate the area under each quadratic function using the standard formula for the area under a parabolic segment, which is: (h/3)(f(x0) + 4f(x1) + f(x2)).
  4. Sum the areas calculated in step 3 for all subintervals to approximate the definite integral.

The objective is to use the approximation repeatedly to compute the area estimation.

How to use Simpson’s rule?

When applying Simpson's Rule, consider the following:

  1. Divide the graph into an even number of uniform subdivisions to approximate the entire area.
  2. Ensure that the half-widths (Δx) are consistent.
  3. Sub-intervals (n) should always be even for the 1-4-1 approximation.


Estimate the definite integral of the function f(x) = x^2 + 4 over the interval [1, 4] using Simpson's Rule with n = 6 sub-intervals.


Step 1: Calculate Δx (half-width):

Δx = (b - a) / n

Δx = (4 - 1) / 6

Δx = 0.5

Step 2: Divide the interval [1, 4] into n = 6 sub-intervals with length Δx = 0.5:

x0= 1, x1 = 1.5, x2 = 2, x3 = 2.5, x4 = 3, x5 = 3.5, x6 = 4

Step 3: Evaluate the function at these endpoints:

f(x0) = f(1) = 1^2 + 4 = 5

4f(x1) = 4f(1.5) = 4(1.5^2 + 4) = 25

2f(x2) = 2f(2) = 2 (2^2 + 4) = 16

4f(x3) = 4f(2.5) = 4(2.5^2 + 4) = 41

2f(x4) = 2f(3) = 2(3^2 + 4) = 26

4f(x5) = 4f(3.5) = 4(3.5^2 + 4) = 65

f(x6) = f(4) = 4^2 + 4 = 20

Step 4: Add these values:

= 5 + 25 + 16 + 41 + 26 + 65 + 20

= 198

Step 5: Multiply with Δx/3:

= 0.5/3 (198)

= 33

The estimated definite integral using Simpson's Rule is 33.

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