Numerical Integration Simpson's & Trapezoidal Cheat Sheet

A printable reference covering trapezoidal rule, Simpson's rule, step size, error bounds, and composite numerical integration for grades 11-12.

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Numerical integration estimates definite integrals when an antiderivative is hard or impossible to find exactly. This cheat sheet covers the trapezoidal rule and Simpson's rule, two common methods for approximating area under a curve. Students need these methods for calculator work, data-based problems, and real-world models where only function values are available. The trapezoidal rule approximates the region using straight-line segments, while Simpson's rule uses parabolic arcs for a usually more accurate estimate. Both methods divide the interval $[a,b]$ into $n$ equal subintervals with step size $\Delta x = \frac{b-a}{n}$ . Simpson's rule requires $n$ to be even, and both methods use weighted sums of function values. Error bounds depend on higher derivatives, so smoother functions usually give better approximations.

Key Facts

For $n$ equal subintervals on $[a,b]$ , the step size is $\Delta x = \frac{b-a}{n}$ .
The partition points are $x_i = a + i\Delta x$ for $i = 0,1,2,\dots,n$ .
The composite trapezoidal rule is $T_n = \frac{\Delta x}{2}\left[f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n)\right]$ .
The composite Simpson's rule is $S_n = \frac{\Delta x}{3}\left[f(x_0) + 4\sum_{\text{odd } i} f(x_i) + 2\sum_{\text{even } i,\ i\ne 0,n} f(x_i) + f(x_n)\right]$ .
Simpson's rule requires an even number of subintervals, so $n$ must be even.
If $f$ is concave up on $[a,b]$ , the trapezoidal rule usually overestimates $\int_a^b f(x)\,dx$ .
If $f$ is concave down on $[a,b]$ , the trapezoidal rule usually underestimates $\int_a^b f(x)\,dx$ .
A common trapezoidal error bound is $|E_T| \le \frac{K(b-a)^3}{12n^2}$ when $|f''(x)| \le K$ on $[a,b]$ .

Vocabulary

Numerical integration: A method for estimating a definite integral using function values instead of finding an exact antiderivative.
Trapezoidal rule: An approximation method that estimates area by connecting points on the graph with straight line segments to form trapezoids.
Simpson's rule: An approximation method that estimates area by fitting parabolas through groups of three points.
Step size: The width of each equal subinterval, given by $\Delta x = \frac{b-a}{n}$ .
Subinterval: One smaller interval created when $[a,b]$ is divided into equal parts for approximation.
Error bound: A maximum possible difference between the exact integral and a numerical approximation.

Common Mistakes to Avoid

Using Simpson's rule with an odd value of $n$ is wrong because Simpson's rule works in pairs of subintervals and requires $n$ to be even.
Forgetting endpoint weights changes the approximation because $f(x_0)$ and $f(x_n)$ are not doubled in the trapezoidal rule or given alternating Simpson weights.
Using $\Delta x = \frac{b-a}{n+1}$ is wrong because $n$ counts subintervals, not the number of listed points.
Mixing Simpson weights in the wrong order gives an incorrect sum because the interior weights must alternate $4,2,4,2,\dots,4$ .
Rounding too early can make the final estimate less accurate, so keep several decimal places until the final answer.

Practice Questions

1 Use the trapezoidal rule with $n=4$ to approximate $\int_0^2 (x^2+1)\,dx$ .
2 Use Simpson's rule with $n=4$ to approximate $\int_0^2 (x^2+1)\,dx$ .
3 For $f(x)=\sin x$ on $[0,\pi]$ , find $\Delta x$ when $n=6$ and list the partition points $x_0$ through $x_6$ .
4 Explain why Simpson's rule often gives a more accurate approximation than the trapezoidal rule for smooth curves.

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