The Central Idea
Numerical integration approximates
\[ \int_a^b f(x)\,dx \]using a finite collection of function values.
The general strategy is:
- divide the interval into subintervals;
- approximate the function on each subinterval;
- compute the corresponding local area or accumulation;
- add the local contributions.
Different numerical methods use different local approximations.
Why Numerical Integration Is Needed
Numerical integration is useful when:
- an antiderivative is difficult to find;
- no elementary antiderivative exists;
- the function is available only as a table of measured values;
- a computer or calculator is being used for approximation;
- a quick estimate is more useful than an exact symbolic expression.
Equal Subintervals
Divide
\[ [a,b] \]into \(n\) equal subintervals.
The width is
\[ \boxed{\Delta x=\frac{b-a}{n}}. \]The partition points are
\[ x_i=a+i\Delta x. \]Left Endpoint Rule
The left endpoint rule uses the left endpoint of each subinterval:
\[ \boxed{ L_n= \Delta x \left[ f(x_0)+f(x_1)+\cdots+f(x_{n-1}) \right] }. \]This is the same left Riemann sum introduced earlier.
Right Endpoint Rule
The right endpoint rule uses the right endpoint of each subinterval:
\[ \boxed{ R_n= \Delta x \left[ f(x_1)+f(x_2)+\cdots+f(x_n) \right] }. \]Midpoint Rule
The midpoint rule evaluates the function at the center of each subinterval.
If
\[ m_i=\frac{x_{i-1}+x_i}{2}, \]then
\[ \boxed{ M_n= \Delta x \sum_{i=1}^{n}f(m_i) }. \]Midpoint estimates are often much more accurate than left or right sums using the same number of subintervals.
The Trapezoidal Rule
Instead of approximating the graph by horizontal rectangles, the trapezoidal rule connects neighboring data points with straight line segments.
Each subinterval contributes the area of a trapezoid.
\[ \boxed{ \text{trapezoid area} = \frac12 (\text{left height}+\text{right height}) (\text{width}) }. \]Composite Trapezoidal Rule
For \(n\) equal subintervals,
\[ \boxed{ T_n= \frac{\Delta x}{2} \left[ f(x_0) +2f(x_1) +2f(x_2) +\cdots +2f(x_{n-1}) +f(x_n) \right] }. \]The endpoint values appear once, while interior values appear twice.
The Trapezoidal Rule as an Average
For equal subintervals,
\[ \boxed{ T_n=\frac{L_n+R_n}{2} }. \]This provides a useful connection between rectangle approximations and trapezoidal approximation.
Simpson’s Rule
Simpson’s Rule improves the local approximation further by fitting quadratic curves rather than horizontal or straight-line pieces.
It is usually substantially more accurate for smooth functions.
The standard composite Simpson’s Rule requires an even number of subintervals.
Composite Simpson’s Rule
If \(n\) is even and
\[ \Delta x=\frac{b-a}{n}, \]then
\[ \boxed{ S_n= \frac{\Delta x}{3} \left[ f(x_0) +4f(x_1) +2f(x_2) +4f(x_3) +\cdots +2f(x_{n-2}) +4f(x_{n-1}) +f(x_n) \right] }. \]The interior weights alternate:
\[ \boxed{4,2,4,2,\ldots,4}. \]Remembering the Simpson Pattern
For \(n\) even, the coefficient pattern is
\[ \boxed{ 1,\ 4,\ 2,\ 4,\ 2,\ \ldots,\ 4,\ 1 }. \]Then multiply the weighted sum by
\[ \boxed{\frac{\Delta x}{3}}. \]Example: Approximate \(\int_0^2 x^2\,dx\)
Use \(n=4\) equal subintervals.
The width is
\[ \Delta x=\frac{2-0}{4}=\frac12. \]The partition points are
\[ 0,\quad\frac12,\quad1,\quad\frac32,\quad2. \]Left Endpoint Approximation
\[ L_4= \frac12 \left[ 0^2+ \left(\frac12\right)^2+ 1^2+ \left(\frac32\right)^2 \right]. \]Therefore,
\[ L_4= \frac12 \left[ 0+\frac14+1+\frac94 \right]. \] \[ \boxed{L_4=\frac74=1.75}. \]Right Endpoint Approximation
\[ R_4= \frac12 \left[ \left(\frac12\right)^2+ 1^2+ \left(\frac32\right)^2+ 2^2 \right]. \]Thus,
\[ \boxed{R_4=\frac{15}{4}=3.75}. \]Midpoint Approximation
The midpoints are
\[ \frac14,\quad\frac34,\quad\frac54,\quad\frac74. \]Therefore,
\[ M_4= \frac12 \left[ \left(\frac14\right)^2+ \left(\frac34\right)^2+ \left(\frac54\right)^2+ \left(\frac74\right)^2 \right]. \]Hence,
\[ \boxed{M_4=\frac{21}{8}=2.625}. \]Trapezoidal Approximation
Since
\[ T_4=\frac{L_4+R_4}{2}, \]we have
\[ T_4= \frac{1.75+3.75}{2}. \]Therefore,
\[ \boxed{T_4=2.75}. \]Simpson Approximation
Use
\[ S_4= \frac{\Delta x}{3} \left[ f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+f(x_4) \right]. \]Thus,
\[ S_4= \frac{1/2}{3} \left[ 0+ 4\left(\frac14\right)+ 2(1)+ 4\left(\frac94\right)+ 4 \right]. \]The weighted sum is
\[ 0+1+2+9+4=16. \]Hence,
\[ \boxed{ S_4=\frac16(16)=\frac83 }. \]In this case Simpson’s Rule is exact because \(x^2\) is quadratic.
Comparing the Estimates
The exact value is
\[ \int_0^2x^2\,dx=\frac83\approx2.667. \]Our estimates were
\[ L_4=1.75, \] \[ M_4=2.625, \] \[ T_4=2.75, \]and
\[ S_4=\frac83. \]This illustrates how different approximation rules can have very different accuracy.
Numerical Integration from a Table
Numerical methods are especially important when the function is known only through measurements.
| \(x\) | \(f(x)\) |
|---|---|
| 0 | 2 |
| 1 | 3 |
| 2 | 5 |
| 3 | 4 |
| 4 | 6 |
Since the spacing is \(1\), these values can be used directly in left, right, trapezoidal, and Simpson approximations.
Example: Trapezoidal Rule from Tabular Data
Using the table above,
\[ T_4= \frac12 \left[ 2+2(3)+2(5)+2(4)+6 \right]. \]Therefore,
\[ T_4= \frac12(32). \]Hence,
\[ \boxed{T_4=16}. \]Example: Simpson’s Rule from the Same Table
Since \(n=4\) is even,
\[ S_4= \frac13 \left[ 2+4(3)+2(5)+4(4)+6 \right]. \]Thus,
\[ S_4= \frac13(46). \]Therefore,
\[ \boxed{S_4=\frac{46}{3}\approx15.333}. \]Unequal Subintervals
The simple composite formulas above assume equal spacing.
With unequal spacing, the trapezoidal rule can still be applied interval-by-interval:
\[ \boxed{ T = \sum_{i=1}^{n} \frac{f(x_{i-1})+f(x_i)}{2} (x_i-x_{i-1}) }. \]Each trapezoid uses its own width.
Approximation Error
Numerical integration replaces a curved function with a simpler local model.
The difference between the approximation and the true integral is the numerical integration error.
In general, error becomes smaller as the partition is refined.
Monotonicity and Endpoint Errors
If \(f\) is increasing and positive:
- left sums tend to underestimate;
- right sums tend to overestimate.
If \(f\) is decreasing and positive, the pattern reverses.
Concavity and the Trapezoidal Rule
If \(f\) is concave up, secant lines tend to lie above the graph.
Therefore the trapezoidal rule tends to overestimate.
If \(f\) is concave down, secant lines tend to lie below the graph, so the trapezoidal rule tends to underestimate.
Concavity and the Midpoint Rule
For smooth functions, midpoint behavior typically goes in the opposite direction from trapezoidal behavior.
- concave up: midpoint tends to underestimate;
- concave down: midpoint tends to overestimate.
Trapezoidal Error Bound
If
\[ |f''(x)|\leq K \]on \([a,b]\), then
\[ \boxed{ |E_T| \leq \frac{K(b-a)^3}{12n^2} }. \]This shows that trapezoidal error decreases on the order of \(1/n^2\).
Midpoint Error Bound
If
\[ |f''(x)|\leq K, \]then
\[ \boxed{ |E_M| \leq \frac{K(b-a)^3}{24n^2} }. \]The midpoint bound has half the constant of the trapezoidal bound.
Simpson Error Bound
If \(n\) is even and
\[ |f^{(4)}(x)|\leq K \]on \([a,b]\), then
\[ \boxed{ |E_S| \leq \frac{K(b-a)^5}{180n^4} }. \]Simpson error decreases on the order of \(1/n^4\), which explains its strong performance for smooth functions.
Exactness for Certain Functions
Numerical methods are approximations in general, but some are exact for particular classes of functions.
- the trapezoidal rule is exact for linear functions;
- the midpoint rule is exact for linear functions;
- Simpson’s Rule is exact for all polynomials of degree at most \(3\).
Choosing a Numerical Method
A practical hierarchy is:
- use left or right sums when only endpoint-style estimates are needed;
- use midpoint for a simple but often strong approximation;
- use trapezoidal when tabular endpoint data are natural;
- use Simpson’s Rule when the spacing is equal, \(n\) is even, and higher accuracy is desired.
Data Quality Matters
Numerical integration cannot recover information that was never measured.
Sparse, noisy, or irregularly spaced data can limit the reliability of an integral estimate.
A mathematically precise quadrature formula does not eliminate uncertainty in the underlying measurements.
Units in Numerical Integration
Each numerical term has the same units as
\[ f(x)\Delta x. \]For example, if \(v(t)\) is in meters per second and \(\Delta t\) is in seconds, then each contribution is in meters.
The numerical integral therefore approximates displacement.
Example: Velocity Data
| \(t\) (s) | \(v(t)\) (m/s) |
|---|---|
| 0 | 4 |
| 2 | 7 |
| 4 | 9 |
| 6 | 8 |
Using the trapezoidal rule with width \(2\),
\[ T_3= \frac{2}{2} \left[ 4+2(7)+2(9)+8 \right]. \]Thus,
\[ \boxed{T_3=44\text{ meters}}. \]Refining the Partition
If the number of equal subintervals is doubled, the width is halved:
\[ \Delta x=\frac{b-a}{n}. \]Smaller widths usually improve the local representation of the function.
Comparing estimates for \(n\), \(2n\), and \(4n\) is a practical way to assess numerical stability.
Common Error: Confusing Data Points with Subintervals
If a table contains \(n+1\) equally spaced data points, then there are \(n\) subintervals.
For example, five data points create four subintervals.
Common Error: Using Simpson’s Rule with Odd \(n\)
The standard composite Simpson’s Rule requires
\[ \boxed{n\text{ even}}. \]If \(n\) is odd, adjust the partition or use another method.
Common Error: Losing the Simpson Weights
The pattern must be
\[ \boxed{1,4,2,4,2,\ldots,4,1}. \]The two endpoints receive coefficient \(1\), odd-indexed interior values receive \(4\), and even-indexed interior values receive \(2\).
Common Error: Forgetting the Width Factor
Every numerical integration rule contains the subinterval width.
A weighted sum of function values alone does not yet have the correct accumulated scale or units.
Common Error: Applying Equal-Spacing Formulas to Unequal Data
Before using a composite formula, verify that the \(x\)-values are equally spaced.
For irregular spacing, compute contributions interval by interval unless a method specifically designed for nonuniform spacing is being used.
A Reliable Numerical Integration Procedure
- Identify the interval and available data.
- Determine the number of subintervals.
- Check whether the spacing is uniform.
- Choose an appropriate rule.
- Write the formula before substituting numbers.
- Apply the correct weights.
- Multiply by the width factor.
- Carry units through the calculation.
- Estimate whether the result should be high or low when possible.
- Refine the partition if greater accuracy is needed.
Practice 1: Width
Divide \([0,8]\) into \(4\) equal subintervals. Find \(\Delta x\).
Practice 2: Left Rule
Approximate
\[ \int_0^4(x+1)\,dx \]using \(4\) left-endpoint rectangles.
Practice 3: Right Rule
Approximate the same integral using \(4\) right-endpoint rectangles.
Practice 4: Midpoint Rule
Approximate
\[ \int_0^2x^2\,dx \]using \(2\) midpoint subintervals.
Practice 5: Trapezoidal Rule
Use \(n=2\) to approximate
\[ \int_0^2x^2\,dx \]with the trapezoidal rule.
Practice 6: Simpson’s Rule
Use \(n=2\) to approximate
\[ \int_0^2x^2\,dx \]with Simpson’s Rule.
Practice 7: Table and Trapezoids
| \(x\) | \(f(x)\) |
|---|---|
| 0 | 1 |
| 1 | 2 |
| 2 | 5 |
| 3 | 4 |
Use the trapezoidal rule on \([0,3]\).
Practice 8: Simpson Weights
Write the coefficient pattern for Simpson’s Rule when \(n=6\).
Practice 9: Over or Under?
If \(f\) is positive, increasing, and concave up, classify the likely direction of error for:
- left endpoint rule;
- right endpoint rule;
- trapezoidal rule;
- midpoint rule.
Practice 10: Simpson Eligibility
Can standard composite Simpson’s Rule be used directly with \(n=5\)?
Practice 11: Units
A rate is measured in gallons per hour and integrated numerically with respect to hours. What are the units of the resulting estimate?
Practice 12: Irregular Spacing
Data are recorded at
\[ x=0,\ 1,\ 3,\ 6. \]Can the standard equal-spacing composite trapezoidal formula be used directly?
Practice Answers
Practice 1
\[ \Delta x=\frac{8-0}{4}=\boxed{2}. \]Practice 2
Here \(\Delta x=1\), and the left endpoints are \(0,1,2,3\).
\[ L_4=(1+2+3+4)(1)=\boxed{10}. \]Practice 3
The right endpoints are \(1,2,3,4\).
\[ R_4=(2+3+4+5)(1)=\boxed{14}. \]Practice 4
The width is \(1\), and the midpoints are \(1/2\) and \(3/2\).
\[ M_2= \left[ \left(\frac12\right)^2+ \left(\frac32\right)^2 \right](1) = \boxed{\frac52}. \]Practice 5
\[ T_2= \frac12 \left[ 0^2+2(1^2)+2^2 \right] = \boxed{3}. \]Practice 6
\[ S_2= \frac13 \left[ 0^2+4(1^2)+2^2 \right] = \frac13(8). \]Therefore,
\[ \boxed{\frac83}. \]Practice 7
\[ T_3= \frac12 [1+2(2)+2(5)+4]. \] \[ =\frac12(19). \]Hence,
\[ \boxed{9.5}. \]Practice 8
\[ \boxed{1,\ 4,\ 2,\ 4,\ 2,\ 4,\ 1}. \]Practice 9
- left endpoint: underestimate;
- right endpoint: overestimate;
- trapezoidal: overestimate because the graph is concave up;
- midpoint: underestimate because the graph is concave up.
Practice 10
No. Standard composite Simpson’s Rule requires an even number of subintervals.
Practice 11
The units are
\[ \frac{\text{gallons}}{\text{hour}} \cdot \text{hours} = \boxed{\text{gallons}}. \]Practice 12
No. The spacing is not uniform.
Use interval-by-interval trapezoids or another method suitable for nonuniform spacing.
What to Remember
Numerical integration estimates
\[ \boxed{\int_a^b f(x)\,dx} \]using finitely many function values.
For equal subintervals,
\[ \boxed{\Delta x=\frac{b-a}{n}}. \]The main rules are:
\[ \boxed{ L_n,\quad R_n,\quad M_n,\quad T_n,\quad S_n }. \]The trapezoidal rule uses straight-line approximations, while Simpson’s Rule uses quadratic approximations.
As a practical principle:
use the simplest method that matches the available data and the accuracy you need, then refine the partition when greater precision is required.
Where We Go Next
Numerical integration gives practical approximations to accumulated quantities. The next applications of integration use those accumulated quantities to compute area, volume, arc length, and work.