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Simpson's Rule (Single Strip)

Estimate the area under a curve using a parabola.

Understand the formulaSee the free derivationOpen the full walkthrough

A \approx \frac{h}{3} (y_{0} + 4 y_{1} + y_{2})

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Core idea

Overview

Simpson's Rule is a numerical integration technique that approximates the area under a curve by fitting a quadratic parabola through three evenly spaced points. It belongs to the Newton-Cotes formulas and provides a more accurate estimation than the Trapezoidal Rule for functions that are reasonably smooth.

When to use: Apply this rule when you need to calculate the definite integral of a function using discrete data points or when the antiderivative is difficult to find. It requires three points (the start, midpoint, and end) spanning two equal sub-intervals of width 'h'.

Why it matters: This rule is a fundamental tool in engineering and physics for approximating work, fluid pressure, and centers of mass from experimental data. It strikes a balance between computational simplicity and high-order accuracy, making it a standard algorithm in scientific software.

Symbols

Variables

h = Step Size, y_0 = First Height, y_1 = Mid Height, y_2 = Last Height, A = Area

h

Step Size

V a r iab l e

y_{0}

First Height

V a r iab l e

y_{1}

Mid Height

V a r iab l e

y_{2}

Last Height

V a r iab l e

A

Area

V a r iab l e

Walkthrough

Derivation

Understanding Simpson's Rule

Simpson’s rule approximates an integral by fitting parabolas through points, typically giving higher accuracy than the trapezium rule for smooth curves.

The interval is split into an even number of strips n.
Strip width is $h = (b - a) / n$ .

Single Parabolic Arch (Two Strips):

This weighting integrates a quadratic exactly when the curve is a parabola through the three points.

Area \approx \frac{h}{3} (y_{0} + 4 y_{1} + y_{2})

Extend Across the Whole Interval:

Add the parabolic-arch contributions: endpoints counted once, odd ordinates weighted 4, even ordinates weighted 2.

\int_{a}^{b} f (x) d x \approx \frac{h}{3} [y_{0} + y_{n} + 4 (y_{1} + y_{3} + \dots) + 2 (y_{2} + y_{4} + \dots)]

Result

\int_{a}^{b} f (x) d x \approx \frac{h}{3} [y_{0} + y_{n} + 4 (y_{1} + y_{3} + \dots) + 2 (y_{2} + y_{4} + \dots)]

Source: Edexcel Further Mathematics — Core Pure (Numerical Methods)

Visual intuition

Graph

The graph represents a quadratic relationship where the area A is a parabolic function of the independent variable. Because the formula calculates the area under a curve using a quadratic approximation, the resulting plot forms a parabola that accounts for the curvature of the function over the interval.

Graph type: parabolic

Why it behaves this way

Intuition

Imagine fitting a smooth parabolic arc through three points on a curve - the start, middle, and end - and then calculating the area under that specific parabola as an estimate for the actual area under the original

Approximation of the definite integral or area under the curve

The calculated total accumulation of the function's value over the specified interval.

Width of each of the two equal sub-intervals

Determines the 'step size' for the approximation; smaller 'h' means a narrower segment and generally a more refined estimate.

y_{0}

Function value at the start of the interval (x_0)

The height of the curve at the very beginning of the segment being approximated.

y_{1}

Function value at the midpoint of the interval (x_1)

The height of the curve exactly halfway across the segment; crucial for defining the curvature of the approximating parabola.

y_{2}

Function value at the end of the interval (x_2)

The height of the curve at the very end of the segment being approximated.

Signs and relationships

h/3: The division by 3 is a weighting factor derived from the integration of the quadratic polynomial used to approximate the curve. It ensures the correct scaling of the area based on the interval width 'h'.
4y_1: The midpoint function value ( $y_{1}$ ) is weighted four times more heavily than the endpoints ( $y_{0}$ , $y_{2}$ ). This higher weighting reflects its greater influence on the curvature of the approximating parabola and its central

Free study cues

Insight

Canonical usage

The unit of the approximated area or integral value (A) is the product of the unit of the interval width (h) and the unit of the function values ( $y_{i}$ ).

Common confusion

Students may use inconsistent units for `h` and ` $y_{i}$ `, leading to an incorrect or non-standard unit for the approximated area `A`.

Unit systems

$h$ length unit (e.g., m, ft) · Represents the width of each sub-interval. Must be in consistent units with the domain of the function.

$y_{0}, y_{1}, y_{2}$ unit of the function being integrated · Represent the function values at specific points. Must be in consistent units with the range of the function.

$A$ product of h's unit and y's unit (e.g., m^2, J) · The approximated area or integral value. Its unit reflects the physical quantity being integrated.

One free problem

Practice Problem

A civil engineer measures the depth of a river cross-section at three points spaced 6 meters apart. The recorded depths are 2 meters, 5 meters, and 2 meters. Calculate the approximate area of the cross-section using Simpson's Rule.

Step Size6

First Height2

Mid Height5

Last Height2

Solve for: $A$

Hint: Plug the values directly into the formula A ≈ (h/3)(y0 + 4y1 + y2).

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Naval architecture (ship stability).

Study smarter

Tips

Ensure the interval width 'h' is constant between the three y-coordinates.
The coefficient for the middle coordinate (y1) is always 4, while the endpoints are 1.
Remember that 'h' is half the total width of the area being measured.

Avoid these traps

Common Mistakes

Using h/2 instead of h/3.
Weights order.

Common questions

Frequently Asked Questions

Simpson’s rule approximates an integral by fitting parabolas through points, typically giving higher accuracy than the trapezium rule for smooth curves.

Apply this rule when you need to calculate the definite integral of a function using discrete data points or when the antiderivative is difficult to find. It requires three points (the start, midpoint, and end) spanning two equal sub-intervals of width 'h'.

This rule is a fundamental tool in engineering and physics for approximating work, fluid pressure, and centers of mass from experimental data. It strikes a balance between computational simplicity and high-order accuracy, making it a standard algorithm in scientific software.

Using h/2 instead of h/3. Weights order.