Key takeaways
- Simpson's Rule: area ≈ (Δx ÷ 3) × [f(x₀) + 4f(x₁) + 2f(x₂) + … + 4f(xₙ₋₁) + f(xₙ)].
- The number of strips N must be even, because each parabola covers two strips.
- The weights go 1, 4, 2, 4, …, 2, 4, 1. The ends count once.
- It is exact for any polynomial up to degree 3, such as x³.
- The #1 mistake is dividing by 2 (the trapezoidal rule) instead of 3.
In this guide
What is Simpson's Rule?
A definite integral is the area under a curve between two x-values, a and b. Sometimes you cannot find that area with a formula, or you only have measured data points. Then you approximate it: you get a very close estimate instead of an exact answer.
Picture the region cut into thin vertical strips. The trapezoidal rule joins the tops of the strips with straight lines. Simpson's Rule does something smarter. It takes the strips two at a time and draws a small parabola (a U-shaped curve) through the three points on top. Curves hug curves much better than straight lines do, so the error shrinks fast.
Why does Simpson's Rule matter?
Many real integrals have no neat answer. The bell curve used in statistics is a famous example. And in real life you often don't have a formula at all, just readings: the width of a river every 2 metres, or the speed of a car every second. Simpson's Rule turns those readings into an area, a volume or a distance.
It is also surprisingly accurate for very little work. Look at the area under y = x⁴ from 0 to 2. The exact answer is 6.4.
| Method (4 strips) | Estimate | Error |
|---|---|---|
| Trapezoidal rule | 7.0625 | 0.6625 |
| Simpson's Rule | 6.4167 | 0.0167 |
Same four strips, same five points, but Simpson's Rule is about 40 times closer to the truth.
The formula
| Symbol | What it means | Example |
|---|---|---|
| a, b | Start and end of the interval | 0 and 2 |
| N | Number of strips (must be even) | 4 |
| Δx | Width of one strip: (b − a) ÷ N | 0.5 |
| x₀ … xₙ | The points a, a + Δx, a + 2Δx, … up to b | 0, 0.5, 1, 1.5, 2 |
| f(x) | Height of the curve at x | f(1) = 1 |
The weights follow the pattern 1, 4, 2, 4, 2, …, 4, 1. The two ends count once. Odd-numbered points (x₁, x₃, …) count 4 times. Even-numbered inside points (x₂, x₄, …) count 2 times.
This calculator works with curves of the form f(x) = xᵖ, where you choose the power p. It also shows the exact answer, xᵖ⁺¹ ÷ (p + 1) between a and b, and the error, so you can see how good the estimate is. If you type an odd N, it adds 1 to make it even.
How to use Simpson's Rule step by step
- 1Choose an even N. Then find the strip width
Δx = (b − a) ÷ N. - 2List the x-values. Start at a and add Δx each time until you reach b.
- 3Find the heights. Work out f(x) at every point.
- 4Apply the weights. Multiply by 1, 4, 2, 4, …, 4, 1 and add everything up.
- 5Multiply by Δx ÷ 3. That number is your estimate of the area.
Now try your own curve. Change the power, the limits or N below and compare the estimate with the exact answer.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Using an odd N. Simpson's Rule needs pairs of strips. With N = 5 the 1-4-2-4-1 pattern breaks.
- Mixing up the weights. Odd positions (x₁, x₃, …) get 4. Even inside positions (x₂, x₄, …) get 2. The two ends get 1.
- Counting points instead of strips. N = 4 strips means 5 points, from x₀ to x₄.
- Expecting an exact answer every time. It is exact for polynomials up to degree 3, but only approximate for x⁴ and beyond.
Tips and tricks
- Quick check: your weights should add up to 3N. For N = 4:
1 + 4 + 2 + 4 + 1 = 12 = 3 × 4. - Compare methods: run the same curve through the Riemann sum calculator to see how much better parabolas are than rectangles.
- Know the true answer? Check it with the definite integral calculator.
Where you'll use it in real life
- Engineering: the area of a river's cross-section from depth readings taken at equal spacing.
- Physics: distance travelled from a log of speed readings, since distance is the area under the speed graph.
- Shipbuilding: naval architects have long used Simpson's Rule to find the areas and volumes of curved hulls.
- Computing: calculators and software use rules like this to evaluate integrals that have no formula.
Quick summary
Simpson's Rule splits the area into an even number of strips, fits parabolas through each pair, and combines the heights with the weights 1, 4, 2, 4, …, 4, 1 times Δx ÷ 3. It is far more accurate than straight-line methods for the same effort. Use the calculator above to compare the estimate with the exact answer for any power of x.
Worked example
Your teacher asks you to estimate the area under y = x⁴ from x = 0 to x = 2 using Simpson's Rule with 4 strips, then compare it with the exact value. So p = 4, a = 0, b = 2 and N = 4, which gives Δx = 0.5.
- 01∫[0 to 2] x^4 dx using Simpson's Rule, N=4
- 02Δx = (2−0)/4 = 0.5
- 03S = (Δx/3)[f(x₀)+4f(x₁)+2f(x₂)+…+4f(xₙ₋₁)+f(xₙ)]
- 04≈ 6.4167
- 05Exact = 6.4
- 06Error = 0.0167
These numbers are pre-filled in the calculator above.
Frequently asked questions
Why does Simpson's Rule need an even number of intervals?
Each parabola passes through three points and covers two strips. To cover the whole interval with complete pairs, the number of strips must be even.
Is Simpson's Rule more accurate than the trapezoidal rule?
For smooth curves, usually yes. Its error shrinks roughly with Δx⁴, while the trapezoidal rule's error shrinks with Δx², so halving the strip width helps Simpson's Rule much more.
When is Simpson's Rule exact?
It gives the exact answer for any polynomial of degree 3 or less. For example, the integral of x³ from 0 to 2 is 4, and Simpson's Rule with N = 2 also gives exactly 4.
Why is it sometimes called Simpson's 1/3 rule?
Because of the Δx ÷ 3 factor in front. There is also a Simpson's 3/8 rule, which fits cubic curves through groups of three strips.
