Trapezoidal Rule - Wikipedia

Numerical integration method This article is about a rule for approximating integrals. For the trapezoidal rule used for initial value problems, see Trapezoidal rule (differential equations) and Heun's method.

In calculus, the trapezoidal rule (informally trapezoid rule; or in British English trapezium rule)[a] is a technique for numerical integration, i.e. approximating the definite integral: ∫ a b f ( x ) d x . {\displaystyle \int _{a}^{b}f(x)\,dx.}

The trapezoidal rule works by approximating the region under the graph of the function f ( x ) {\displaystyle f(x)} as a trapezoid and calculating its area. This is easily calculated by noting that the area of the region is made up of a rectangle with width ( b − a ) {\displaystyle (b-a)} and height f ( a ) {\displaystyle f(a)} , and a triangle of width ( b − a ) {\displaystyle (b-a)} and height f ( b ) − f ( a ) {\displaystyle f(b)-f(a)} .

Letting A r {\displaystyle A_{r}} denote the area of the rectangle and A t {\displaystyle A_{\text{t}}} the area of the triangle, it follows that A r = ( b − a ) ⋅ f ( a ) , A t = 1 2 ( b − a ) ⋅ [ f ( b ) − f ( a ) ] . {\displaystyle A_{\text{r}}=(b-a)\cdot f(a),\quad A_{\text{t}}={\tfrac {1}{2}}(b-a)\cdot [f(b)-f(a)].}

Therefore, ∫ a b f ( x ) d x ≈ A r + A t = ( b − a ) ⋅ f ( a ) + 1 2 ( b − a ) ⋅ [ f ( b ) − f ( a ) ] = ( b − a ) ⋅ ( f ( a ) + 1 2 f ( b ) − 1 2 f ( a ) ) = ( b − a ) ⋅ ( 1 2 f ( a ) + 1 2 f ( b ) ) = ( b − a ) ⋅ 1 2 [ f ( a ) + f ( b ) ] . {\displaystyle {\begin{aligned}\int _{a}^{b}f(x)\,dx&\approx A_{\text{r}}+A_{\text{t}}\\&=(b-a)\cdot f(a)+{\tfrac {1}{2}}(b-a)\cdot [f(b)-f(a)]\\&=(b-a)\cdot \left(f(a)+{\tfrac {1}{2}}f(b)-{\tfrac {1}{2}}f(a)\right)\\&=(b-a)\cdot \left({\tfrac {1}{2}}f(a)+{\tfrac {1}{2}}f(b)\right)\\&=(b-a)\cdot {\tfrac {1}{2}}[f(a)+f(b)].\end{aligned}}}

The rule can also be derived by replacing the integrand with the equation of the line joining points ( a , f ( a ) ) {\displaystyle {\big (}a,f(a){\big )}} and ( b , f ( b ) ) {\displaystyle {\big (}b,f(b){\big )}} , which using the two point form of the equation of a line, is y = ( x − a ) f ( b ) − f ( a ) b − a + f ( a ) . {\displaystyle y=(x-a)\,{\frac {f(b)-f(a)}{b-a}}+f(a).}

Therefore, ∫ a b f ( x ) d x ≈ ∫ a b ( x − a ) f ( b ) − f ( a ) b − a + f ( a ) d x = [ 1 2 ( x − a ) 2 f ( b ) − f ( a ) b − a + x f ( a ) ] x = a x = b = [ 1 2 ( b − a ) 2 f ( b ) − f ( a ) b − a + b f ( a ) ] − a f ( a ) = 1 2 ( b − a ) [ f ( b ) − f ( a ) ] + ( b − a ) f ( a ) = 1 2 ( b − a ) [ f ( a ) + f ( b ) ] , {\displaystyle {\begin{aligned}\int _{a}^{b}f(x)\,dx&\approx \int _{a}^{b}(x-a)\,{\frac {f(b)-f(a)}{b-a}}+f(a)\,dx\\&=\left[{\frac {1}{2}}(x-a)^{2}\,{\frac {f(b)-f(a)}{b-a}}+xf(a)\right]_{x=a}^{x=b}\\&=\left[{\frac {1}{2}}(b-a)^{2}\,{\frac {f(b)-f(a)}{b-a}}+bf(a)\right]-af(a)\\&={\frac {1}{2}}(b-a)[f(b)-f(a)]+(b-a)f(a)\\&={\frac {1}{2}}(b-a)[f(a)+f(b)],\end{aligned}}} as before.

The integral can be even better approximated by partitioning the integration interval, applying the trapezoidal rule to each subinterval and summing the results. In practice, this "chained" (or "composite") trapezoidal rule is usually what is meant by "integrating with the trapezoidal rule". Let { x k } {\displaystyle \{x_{k}\}} be a partition of [ a , b ] {\displaystyle [a,b]} such that a = x 0 < x 1 < ⋯ < x N − 1 < x N = b , {\displaystyle a=x_{0}<x_{1}<\cdots <x_{N-1}<x_{N}=b,} and Δ x k {\displaystyle \Delta x_{k}} be the length of the k {\displaystyle k} -th subinterval (that is, Δ x k = x k − x k − 1 {\displaystyle \Delta x_{k}=x_{k}-x_{k-1}} ), then ∫ a b f ( x ) d x ≈ ∑ k = 1 N f ( x k − 1 ) + f ( x k ) 2 Δ x k . {\displaystyle \int _{a}^{b}f(x)\,dx\approx \sum _{k=1}^{N}{\frac {f(x_{k-1})+f(x_{k})}{2}}\Delta x_{k}.} The trapezoidal rule may be viewed as the result obtained by averaging the left and right Riemann sums and is sometimes defined this way.

The approximation becomes more accurate as the resolution of the partition increases (that is, for larger N {\displaystyle N} , all Δ x k {\displaystyle \Delta x_{k}} decrease).

When the partition has a regular spacing, as is often the case, that is, when all the Δ x k {\displaystyle \Delta x_{k}} have the same value Δ x , {\displaystyle \Delta x,} the formula can be simplified for calculation efficiency by factoring Δ x {\displaystyle \Delta x} out: ∫ a b f ( x ) d x ≈ Δ x ( f ( x 0 ) + f ( x N ) 2 + ∑ k = 1 N − 1 f ( x k ) ) . {\displaystyle \int _{a}^{b}f(x)\,dx\approx \Delta x\left({\frac {f(x_{0})+f(x_{N})}{2}}+\sum _{k=1}^{N-1}f(x_{k})\right).}

As discussed below, it is also possible to place error bounds on the accuracy of the value of a definite integral estimated using a trapezoidal rule.

History

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A 2016 Science paper reports that the trapezoid rule was in use in Babylon before 50 BCE for integrating the velocity of Jupiter along the ecliptic.[1][2]

Numerical implementation

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Non-uniform grid

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When the grid spacing is non-uniform, one can use the formula ∫ a b f ( x ) d x ≈ ∑ k = 1 N f ( x k − 1 ) + f ( x k ) 2 Δ x k , {\displaystyle \int _{a}^{b}f(x)\,dx\approx \sum _{k=1}^{N}{\frac {f(x_{k-1})+f(x_{k})}{2}}\Delta x_{k},} where Δ x k = x k − x k − 1 , {\displaystyle \Delta x_{k}=x_{k}-x_{k-1},} or more a computationally efficient formula ∫ a b f ( x ) d x ≈ 1 2 ( f ( x 0 ) Δ + 1 x 0 + f ( x N ) Δ − 1 x N + ∑ k = 1 N − 1 f ( x k ) Δ ± 1 x k ) , {\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {1}{2}}{\biggl (}f(x_{0})\Delta _{+1}x_{0}+f(x_{N})\Delta _{-1}x_{N}+\sum _{k=1}^{N-1}f(x_{k})\Delta _{\pm 1}x_{k}{\biggr )},} where Δ + 1 x 0 = x 1 − x 0 , {\displaystyle \Delta _{+1}x_{0}=x_{1}-x_{0},} Δ − 1 x N = x N − x N − 1 , {\displaystyle \Delta _{-1}x_{N}=x_{N}-x_{N-1},} Δ ± 1 x k = x k + 1 − x k − 1 {\displaystyle \Delta _{\pm 1}x_{k}=x_{k+1}-x_{k-1}} are the corresponding forward, backward, and central differences.

Uniform grid

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For a domain partitioned by N {\displaystyle N} equally spaced points, considerable simplification may occur.

Let Δ x = b − a N {\displaystyle \Delta x={\frac {b-a}{N}}} and x k = a + k Δ x {\displaystyle x_{k}=a+k\Delta x} for k = 0 , 1 , … , N . {\displaystyle k=0,1,\ldots ,N.} The approximation to the integral becomes ∫ a b f ( x ) d x ≈ Δ x 2 ∑ k = 1 N [ f ( x k − 1 ) + f ( x k ) ] = Δ x ( 1 2 f ( x 0 ) + 1 2 f ( x N ) + ∑ k = 1 N − 1 f ( x k ) ) . {\displaystyle {\begin{aligned}\int _{a}^{b}f(x)\,dx&\approx {\frac {\Delta x}{2}}\sum _{k=1}^{N}[f(x_{k-1})+f(x_{k})]\\&=\Delta x{\biggl (}{\tfrac {1}{2}}f(x_{0})+{\tfrac {1}{2}}f(x_{N})+\sum _{k=1}^{N-1}f(x_{k}){\biggr )}.\end{aligned}}}

Sometimes this expression is written as Δ x   ∑ ) ′ k = 0 N ⁡ f ( x k ) , {\displaystyle \Delta x\!\!\mathop {\ \sum {\vphantom {\big )}}'} _{k=0}^{N}f(x_{k}),}

where the symbol ⁠ ∑   ′ {\displaystyle \textstyle \sum ~\!\!'} ⁠ indicates that the first and last terms are halved.

Error analysis

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The error of the composite trapezoidal rule is the difference between the value of the integral and the numerical result: E = ∫ a b f ( x ) d x − b − a N [ f ( a ) + f ( b ) 2 + ∑ k = 1 N − 1 f ( a + k b − a N ) ] {\displaystyle {\text{E}}=\int _{a}^{b}f(x)\,dx-{\frac {b-a}{N}}\left[{f(a)+f(b) \over 2}+\sum _{k=1}^{N-1}f\left(a+k{\frac {b-a}{N}}\right)\right]}

There exists a number ξ between a and b, such that[3] E = − ( b − a ) 3 12 N 2 f ″ ( ξ ) {\displaystyle {\text{E}}=-{\frac {(b-a)^{3}}{12N^{2}}}f''(\xi )}

It follows that if the integrand is concave up (and thus has a positive second derivative), then the error is negative and the trapezoidal rule overestimates the true value. This can also be seen from the geometric picture: the trapezoids include all of the area under the curve and extend over it. Similarly, a concave-down function yields an underestimate because area is unaccounted for under the curve, but none is counted above. If the interval of the integral being approximated includes an inflection point, the sign of the error is harder to identify.

An asymptotic error estimate for N → ∞ is given by E = − ( b − a ) 2 12 N 2 [ f ′ ( b ) − f ′ ( a ) ] + O ( N − 3 ) . {\displaystyle {\text{E}}=-{\frac {(b-a)^{2}}{12N^{2}}}{\big [}f'(b)-f'(a){\big ]}+O(N^{-3}).} Further terms in this error estimate are given by the Euler–Maclaurin summation formula.

Several techniques can be used to analyze the error, including:[4]

1. Fourier series
2. Residue calculus
3. Euler–Maclaurin summation formula[5][6]
4. Polynomial interpolation[7]

It is argued that the speed of convergence of the trapezoidal rule reflects and can be used as a definition of classes of smoothness of the functions.[8]

Proof

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First suppose that h = b − a N {\displaystyle h={\frac {b-a}{N}}} and a k = a + ( k − 1 ) h {\displaystyle a_{k}=a+(k-1)h} . Let g k ( t ) = 1 2 t [ f ( a k ) + f ( a k + t ) ] − ∫ a k a k + t f ( x ) d x {\displaystyle g_{k}(t)={\frac {1}{2}}t[f(a_{k})+f(a_{k}+t)]-\int _{a_{k}}^{a_{k}+t}f(x)\,dx} be the function such that | g k ( h ) | {\displaystyle |g_{k}(h)|} is the error of the trapezoidal rule on one of the intervals, [ a k , a k + h ] {\displaystyle [a_{k},a_{k}+h]} . Then d g k d t = 1 2 [ f ( a k ) + f ( a k + t ) ] + 1 2 t ⋅ f ′ ( a k + t ) − f ( a k + t ) , {\displaystyle {dg_{k} \over dt}={1 \over 2}[f(a_{k})+f(a_{k}+t)]+{1 \over 2}t\cdot f'(a_{k}+t)-f(a_{k}+t),} and d 2 g k d t 2 = 1 2 t ⋅ f ″ ( a k + t ) . {\displaystyle {d^{2}g_{k} \over dt^{2}}={1 \over 2}t\cdot f''(a_{k}+t).}

Now suppose that | f ″ ( x ) | ≤ | f ″ ( ξ ) | , {\displaystyle \left|f''(x)\right|\leq \left|f''(\xi )\right|,} which holds if f {\displaystyle f} is sufficiently smooth. It then follows that | f ″ ( a k + t ) | ≤ f ″ ( ξ ) {\displaystyle \left|f''(a_{k}+t)\right|\leq f''(\xi )} which is equivalent to − f ″ ( ξ ) ≤ f ″ ( a k + t ) ≤ f ″ ( ξ ) {\displaystyle -f''(\xi )\leq f''(a_{k}+t)\leq f''(\xi )} , or − f ″ ( ξ ) t 2 ≤ g k ″ ( t ) ≤ f ″ ( ξ ) t 2 . {\displaystyle -{\frac {f''(\xi )t}{2}}\leq g_{k}''(t)\leq {\frac {f''(\xi )t}{2}}.}

Since g k ′ ( 0 ) = 0 {\displaystyle g_{k}'(0)=0} and g k ( 0 ) = 0 {\displaystyle g_{k}(0)=0} , ∫ 0 t g k ″ ( x ) d x = g k ′ ( t ) {\displaystyle \int _{0}^{t}g_{k}''(x)dx=g_{k}'(t)} and ∫ 0 t g k ′ ( x ) d x = g k ( t ) . {\displaystyle \int _{0}^{t}g_{k}'(x)dx=g_{k}(t).}

Using these results, we find − f ″ ( ξ ) t 2 4 ≤ g k ′ ( t ) ≤ f ″ ( ξ ) t 2 4 {\displaystyle -{\frac {f''(\xi )t^{2}}{4}}\leq g_{k}'(t)\leq {\frac {f''(\xi )t^{2}}{4}}} and − f ″ ( ξ ) t 3 12 ≤ g k ( t ) ≤ f ″ ( ξ ) t 3 12 {\displaystyle -{\frac {f''(\xi )t^{3}}{12}}\leq g_{k}(t)\leq {\frac {f''(\xi )t^{3}}{12}}}

Letting t = h {\displaystyle t=h} we find − f ″ ( ξ ) h 3 12 ≤ g k ( h ) ≤ f ″ ( ξ ) h 3 12 . {\displaystyle -{\frac {f''(\xi )h^{3}}{12}}\leq g_{k}(h)\leq {\frac {f''(\xi )h^{3}}{12}}.}

Summing all of the local error terms we find ∑ k = 1 N g k ( h ) = b − a N [ f ( a ) + f ( b ) 2 + ∑ k = 1 N − 1 f ( a + k b − a N ) ] − ∫ a b f ( x ) d x . {\displaystyle \sum _{k=1}^{N}g_{k}(h)={\frac {b-a}{N}}\left[{f(a)+f(b) \over 2}+\sum _{k=1}^{N-1}f\left(a+k{\frac {b-a}{N}}\right)\right]-\int _{a}^{b}f(x)dx.}

But we also have − ∑ k = 1 N f ″ ( ξ ) h 3 12 ≤ ∑ k = 1 N g k ( h ) ≤ ∑ k = 1 N f ″ ( ξ ) h 3 12 {\displaystyle -\sum _{k=1}^{N}{\frac {f''(\xi )h^{3}}{12}}\leq \sum _{k=1}^{N}g_{k}(h)\leq \sum _{k=1}^{N}{\frac {f''(\xi )h^{3}}{12}}} and ∑ k = 1 N f ″ ( ξ ) h 3 12 = f ″ ( ξ ) h 3 N 12 , {\displaystyle \sum _{k=1}^{N}{\frac {f''(\xi )h^{3}}{12}}={\frac {f''(\xi )h^{3}N}{12}},}

so that

− f ″ ( ξ ) h 3 N 12 ≤ b − a N [ f ( a ) + f ( b ) 2 + ∑ k = 1 N − 1 f ( a + k b − a N ) ] − ∫ a b f ( x ) d x ≤ f ″ ( ξ ) h 3 N 12 . {\displaystyle -{\frac {f''(\xi )h^{3}N}{12}}\leq {\frac {b-a}{N}}\left[{f(a)+f(b) \over 2}+\sum _{k=1}^{N-1}f\left(a+k{\frac {b-a}{N}}\right)\right]-\int _{a}^{b}f(x)dx\leq {\frac {f''(\xi )h^{3}N}{12}}.}

Therefore the total error is bounded by

error = ∫ a b f ( x ) d x − b − a N [ f ( a ) + f ( b ) 2 + ∑ k = 1 N − 1 f ( a + k b − a N ) ] = f ″ ( ξ ) h 3 N 12 = f ″ ( ξ ) ( b − a ) 3 12 N 2 . {\displaystyle {\text{error}}=\int _{a}^{b}f(x)\,dx-{\frac {b-a}{N}}\left[{f(a)+f(b) \over 2}+\sum _{k=1}^{N-1}f\left(a+k{\frac {b-a}{N}}\right)\right]={\frac {f''(\xi )h^{3}N}{12}}={\frac {f''(\xi )(b-a)^{3}}{12N^{2}}}.}

Periodic and peak functions

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The trapezoidal rule converges rapidly for periodic functions. This is an easy consequence of the Euler–Maclaurin summation formula, which says that if f {\displaystyle f} is p {\displaystyle p} times continuously differentiable with period T , {\displaystyle T,} then ∑ k = 0 N − 1 f ( k h ) h = ∫ 0 T f ( x ) d x + ∑ k = 1 ⌊ p / 2 ⌋ B 2 k ( 2 k ) ! ( f ( 2 k − 1 ) ( T ) − f ( 2 k − 1 ) ( 0 ) ) − ( − 1 ) p h p ∫ 0 T B ~ p ( x / T ) f ( p ) ( x ) d x , {\displaystyle \sum _{k=0}^{N-1}f(kh)h=\int _{0}^{T}f(x)\,dx+\sum _{k=1}^{\lfloor p/2\rfloor }{\frac {B_{2k}}{(2k)!}}\left(f^{(2k-1)}(T)-f^{(2k-1)}(0)\right)-(-1)^{p}h^{p}\int _{0}^{T}{\tilde {B}}_{p}(x/T)f^{(p)}(x)\,dx,} where h := T / N , {\displaystyle h:=T/N,} and B ~ p {\displaystyle {\tilde {B}}_{p}} is the periodic extension of the p {\displaystyle p} -th Bernoulli polynomial.[9] Due to the periodicity, the derivatives at the endpoint cancel, and we see that the error is O ( h p ) {\displaystyle O(h^{p})} .

A similar effect is available for peak-like functions, such as Gaussian, Exponentially modified Gaussian and other functions with derivatives at integration limits that can be neglected.[10] The evaluation of the full integral of a Gaussian function by trapezoidal rule with 1% accuracy can be made using just 4 points.[11] Simpson's rule requires 1.8 times more points to achieve the same accuracy.[11][12]

"Rough" functions

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For functions that are not in C2, the error bound given above is not applicable. Still, error bounds for such rough functions can be derived, which typically show a slower convergence with the number of function evaluations N {\displaystyle N} than the O ( N − 2 ) {\displaystyle O(N^{-2})} behaviour given above. Interestingly, in this case the trapezoidal rule often has sharper bounds than Simpson's rule for the same number of function evaluations.[13]

Applicability and alternatives

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The trapezoidal rule is one of a family of formulas for numerical integration called Newton–Cotes formulas, of which the midpoint rule is similar to the trapezoid rule. Simpson's rule is another member of the same family, and in general has faster convergence than the trapezoidal rule for functions which are twice continuously differentiable, though not in all specific cases. However, for various classes of rougher functions (ones with weaker smoothness conditions), the trapezoidal rule has faster convergence in general than Simpson's rule.[13]

Moreover, the trapezoidal rule tends to become extremely accurate when periodic functions are integrated over their periods, which can be analyzed in various ways. Convergence usually is exponential or faster.[8][12] A similar effect is available for peak functions.[11][12]

For non-periodic functions, however, methods with unequally spaced points such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally far more accurate; Clenshaw–Curtis quadrature can be viewed as a change of variables to express arbitrary integrals in terms of periodic integrals, at which point the trapezoidal rule can be applied accurately.

Numerical examples

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Approximating the natural logarithm of 3

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Since ∫ 1 3 1 x d x = ln ⁡ 3 − ln ⁡ 1 = ln ⁡ 3 , {\displaystyle \int _{1}^{3}{\frac {1}{x}}\,dx=\ln 3-\ln 1=\ln 3,} we can use the trapezoidal rule to approximate the integral, thereby generating an approximation of ln ⁡ 3 {\displaystyle \ln 3} .

Applying the rule with n = 3 {\displaystyle n=3} segments gives ln ⁡ 3 = ∫ 1 3 1 x d x ≈ 1 3 ( 1 + 6 5 + 6 7 + 1 3 ) = 356 315 ≈ 1.13015873 , {\displaystyle \ln 3=\int _{1}^{3}{\frac {1}{x}}\,dx\approx {\frac {1}{3}}\left(1+{\frac {6}{5}}+{\frac {6}{7}}+{\frac {1}{3}}\right)={\frac {356}{315}}\approx 1.13015873,} which has absolute error of 0.031546 {\displaystyle 0.031546} and a relative error of 2.87148 % {\displaystyle 2.87148\%} .

Applying the rule with n = 6 {\displaystyle n=6} segments gives ln ⁡ 3 = ∫ 1 3 1 x d x ≈ 1 6 ( 1 + 3 2 + 6 5 + 1 + 6 7 + 3 4 + 1 3 ) = 2789 2520 ≈ 1.10674603 , {\displaystyle \ln 3=\int _{1}^{3}{\frac {1}{x}}\,dx\approx {\frac {1}{6}}\left(1+{\frac {3}{2}}+{\frac {6}{5}}+1+{\frac {6}{7}}+{\frac {3}{4}}+{\frac {1}{3}}\right)={\frac {2789}{2520}}\approx 1.10674603,} which has absolute error of 8.13 × 10 − 3 {\displaystyle 8.13\times 10^{-3}} and a relative error of 0.74036 % {\displaystyle 0.74036\%} .

Approximating the integral of a product

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The following integral is given: ∫ 0.1 1.3 5 x e − 2 x d x . {\displaystyle \int _{0.1}^{1.3}5xe^{-2x}\,dx.}

1.  Use the composite trapezoidal rule to estimate the value of this integral. Use three segments.
2.  Find the true error E t {\displaystyle E_{t}} for part (a).
3.  Find the absolute relative true error | ε t | {\displaystyle |\varepsilon _{t}|} for part (a).

Solution

1. The solution using the composite trapezoidal rule with 3 segments is applied as follows.

   ∫ a b f ( x ) d x ≈ b − a 2 n [ f ( a ) + 2 ∑ i = 1 n − 1 f ( a + i h ) + f ( b ) ] . {\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {b-a}{2n}}\left[f(a)+2\sum _{i=1}^{n-1}f(a+ih)+f(b)\right].} n = 3 , a = 0.1 , b = 1.3 , h = b − a n = 1.3 − 0.1 3 = 0.4. {\displaystyle {\begin{aligned}n&=3,\\a&=0.1,\\b&=1.3,\\h&={\frac {b-a}{n}}={\frac {1.3-0.1}{3}}=0.4.\end{aligned}}}

   Using the composite trapezoidal rule formula, ∫ a b f ( x ) d x ≈ b − a 2 n [ f ( a ) + 2 { ∑ i = 1 n − 1 f ( a + i h ) } + f ( b ) ] . {\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {b-a}{2n}}\left[f(a)+2\left\{\sum _{i=1}^{n-1}f(a+ih)\right\}+f(b)\right].}

   I ≈ 1.3 − 0.1 6 [ f ( 0.1 ) + 2 ∑ i = 1 3 − 1 f ( 0.1 + 0.4 i ) + f ( 1.3 ) ] I ≈ 1.3 − 0.1 6 [ f ( 0.1 ) + 2 ∑ i = 1 2 f ( 0.1 + 0.4 i ) + f ( 1.3 ) ] = 0.2 [ f ( 0.1 ) + 2 f ( 0.5 ) + 2 f ( 0.9 ) + f ( 1.3 ) ] = 0.2 [ 5 × 0.1 × e − 2 ( 0.1 ) + 2 ( 5 × 0.5 × e − 2 ( 0.5 ) ) + 2 ( 5 × 0.9 × e − 2 ( 0.9 ) ) + 5 × 1.3 × e − 2 ( 1.3 ) ] = 0.84385. {\displaystyle {\begin{aligned}I&\approx {\frac {1.3-0.1}{6}}\left[f(0.1)+2\sum _{i=1}^{3-1}f(0.1+0.4i)+f(1.3)\right]\\I&\approx {\frac {1.3-0.1}{6}}\left[f(0.1)+2\sum _{i=1}^{2}f(0.1+0.4i)+f(1.3)\right]\\&=0.2[f(0.1)+2f(0.5)+2f(0.9)+f(1.3)]\\&=0.2\left[5\times 0.1\times e^{-2(0.1)}+2(5\times 0.5\times e^{-2(0.5)})+2(5\times 0.9\times e^{-2(0.9)})+5\times 1.3\times e^{-2(1.3)}\right]\\&=0.84385.\end{aligned}}}
2. The exact value of the above integral can be found by integration by parts and is ∫ 0.1 1.3 5 x e − 2 x d x = 0.89387 , {\displaystyle \int _{0.1}^{1.3}5xe^{-2x}\,dx=0.89387,} so the true error is E t = ( true value ) − ( approximate value ) = 0.89387 − 0.84385 = 0.05002. {\displaystyle {\begin{aligned}E_{t}&=({\text{true value}})-({\text{approximate value}})\\&=0.89387-0.84385\\&=0.05002.\end{aligned}}}
3. The absolute relative true error is | ε t | = | true error true value | × 100 % = | 0.05002 0.89387 | × 100 % = 5.5959 % . {\displaystyle {\begin{aligned}|\varepsilon _{t}|&=\left|{\frac {\text{true error}}{\text{true value}}}\right|\times 100\%\\&=\left|{\frac {0.05002}{0.89387}}\right|\times 100\%\\&=5.5959\%.\end{aligned}}}

See also

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• Gaussian quadrature
• Newton–Cotes formulas
• Rectangle method
• Romberg's method
• Simpson's rule
• Clenshaw–Curtis quadrature
• Tai's model
• Volterra integral equation § Numerical solution using trapezoidal rule

Notes

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1. ^ See Trapezoid for more information on terminology.

1. ^ Ossendrijver, Mathieu (Jan 29, 2016). "Ancient Babylonian astronomers calculated Jupiter's position from the area under a time-velocity graph". Science. 351 (6272): 482–484. Bibcode:2016Sci...351..482O. doi:10.1126/science.aad8085. PMID 26823423. S2CID 206644971.
2. ^ "Ancient Babylonians 'first to use geometry'". BBC News. 2016-01-29. Retrieved 2025-02-13.
3. ^ Atkinson 1989, equation (5.1.7).
4. ^ Weideman 2002, p. 23, section 2.
5. ^ Atkinson 1989, equation (5.1.9).
6. ^ Atkinson 1989, p. 285.
7. ^ Burden & Faires 2011, p. 194.
8. ^ a b Rahman & Schmeisser 1990.
9. ^ Kress, Rainer (1998). Numerical Analysis, volume 181 of Graduate Texts in Mathematics. Springer-Verlag.
10. ^ Goodwin, E. T. (1949). "The evaluation of integrals of the form ∫ − ∞ ∞ f ( x ) e − x 2 d x {\displaystyle \textstyle \int _{-\infty }^{\infty }f(x)e^{-x^{2}}\,dx} ". Mathematical Proceedings of the Cambridge Philosophical Society. 45 (2): 241–245. Bibcode:1949PCPS...45..241G. doi:10.1017/S0305004100024786. ISSN 1469-8064.
11. ^ a b c Kalambet, Yuri; Kozmin, Yuri; Samokhin, Andrey (2018). "Comparison of integration rules in the case of very narrow chromatographic peaks". Chemometrics and Intelligent Laboratory Systems. 179: 22–30. doi:10.1016/j.chemolab.2018.06.001. ISSN 0169-7439.
12. ^ a b c Weideman 2002.
13. ^ a b Cruz-Uribe & Neugebauer 2002.

References

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• Atkinson, Kendall E. (1989), An Introduction to Numerical Analysis (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-50023-0
• Rahman, Qazi I.; Schmeisser, Gerhard (December 1990), "Characterization of the speed of convergence of the trapezoidal rule", Numerische Mathematik, 57 (1): 123–138, doi:10.1007/BF01386402, ISSN 0945-3245, S2CID 122245944
• Burden, Richard L.; Faires, J. Douglas (2011), Numerical Analysis (9th ed.), Brooks/Cole
• Weideman, J. A. C. (January 2002), "Numerical Integration of Periodic Functions: A Few Examples", The American Mathematical Monthly, 109 (1): 21–36, doi:10.2307/2695765, JSTOR 2695765
• Cruz-Uribe, D.; Neugebauer, C. J. (2002), "Sharp Error Bounds for the Trapezoidal Rule and Simpson's Rule" (PDF), Journal of Inequalities in Pure and Applied Mathematics, 3 (4)

External links

[edit] The Wikibook A-level Mathematics has a page on the topic of: Trapezium Rule

• I. P. Mysovskikh, Trapezium formula, Encyclopedia of Mathematics, ed. M. Hazewinkel
• Notes on the convergence of trapezoidal-rule quadrature
• An implementation of trapezoidal quadrature provided by Boost.Math

v t e Numerical integration
Newton–Cotes formulas	Trapezoidal rule Simpson's rule Simpson's 3/8 rule Adaptive Simpson's method Boole's rule Romberg's method
Gaussian quadrature	Gauss–Hermite quadrature Gauss–Jacobi quadrature Gauss–Kronrod quadrature formula Gauss–Laguerre quadrature Gauss–Legendre quadrature Chebyshev–Gauss quadrature
Other	Barnes–Hut simulation Bayesian quadrature Clenshaw–Curtis quadrature Filon quadrature Lebedev quadrature Tanh-sinh quadrature
Related	Numerical methods for ordinary differential equations Numerical methods for partial differential equations

v t e Calculus
Precalculus	Binomial theorem Concave function Continuous function Factorial Finite difference Free variables and bound variables Graph of a function Linear function Radian Rolle's theorem Secant Slope Tangent
Limits	Indeterminate form Limit of a function One-sided limit Limit of a sequence Order of approximation (ε, δ)-definition of limit
Differential calculus	Derivative Second derivative Partial derivative Differential Differential operator Mean value theorem Notation Leibniz's notation Newton's notation Rules of differentiation linearity Power Sum Chain L'Hôpital's Product General Leibniz's rule Quotient Other techniques Implicit differentiation Inverse functions and differentiation Logarithmic derivative Related rates Stationary points First derivative test Second derivative test Extreme value theorem Maximum and minimum Further applications Newton's method Taylor's theorem Differential equation Ordinary differential equation Partial differential equation Stochastic differential equation
Integral calculus	Antiderivative Arc length Riemann integral Basic properties Constant of integration Fundamental theorem of calculus Differentiating under the integral sign Integration by parts Integration by substitution trigonometric Euler Tangent half-angle substitution Partial fractions in integration Quadratic integral Trapezoidal rule Volumes Washer method Shell method Integral equation Integro-differential equation
Vector calculus	Derivatives Curl Directional derivative Divergence Gradient Laplacian Basic theorems Line integrals Green's Stokes' Gauss'
Multivariable calculus	Divergence theorem Geometric Hessian matrix Jacobian matrix and determinant Lagrange multiplier Line integral Matrix Multiple integral Partial derivative Surface integral Volume integral Advanced topics Differential forms Exterior derivative Generalized Stokes' theorem Tensor calculus
Sequences and series	Arithmetico-geometric sequence Types of series Alternating Binomial Fourier Geometric Harmonic Infinite Power Maclaurin Taylor Telescoping Tests of convergence Abel's Alternating series Cauchy condensation Direct comparison Dirichlet's Integral Limit comparison Ratio Root Term
Special functionsand numbers	Bernoulli numbers e (mathematical constant) Exponential function Natural logarithm Stirling's approximation
History of calculus	Adequality Brook Taylor Colin Maclaurin Generality of algebra Gottfried Wilhelm Leibniz Infinitesimal Infinitesimal calculus Isaac Newton Fluxion Law of Continuity Leonhard Euler Method of Fluxions The Method of Mechanical Theorems
Lists	Integrals rational functions irrational algebraic functions exponential functions logarithmic functions hyperbolic functions inverse trigonometric functions inverse Secant Secant cubed List of limits List of derivatives
Miscellaneous topics	Complex calculus Contour integral Differential geometry Manifold Curvature of curves of surfaces Tensor Euler–Maclaurin formula Gabriel's horn Integration Bee Proof that 22/7 exceeds π Regiomontanus' angle maximization problem Steinmetz solid