Calculus I · Unit 3A · lesson

Simpson’s Rule

Concept

Learning objectives

Apply Simpson’s Rule with an even number of subintervals and understand why parabolic approximation is unusually accurate.

Simpson's Rule

Explanation

Parabolas can approximate curvature better than lines

Simpson's Rule improves on straight-line approximation by fitting quadratic behavior across pairs of subintervals. This produces the characteristic weights $1,4,2,4,\ldots,2,4,1$ and usually gives high accuracy for smooth functions. The method requires an even number of equal subintervals because each quadratic panel spans two of them.

The formula is easy to misuse if the data spacing or weight pattern is not checked first. Write the table of $x$ -values and weights before multiplying, and verify that the endpoints receive weight one. Simpson's Rule is still an approximation, not a symbolic antiderivative. Its strength is that it captures curvature efficiently, making it useful when function evaluations are available but exact integration is not.

For even $n$ ,

S_n=\frac{\Delta x}{3}\left[f(x_0)+4f(x_1)+2f(x_2)+\cdots+4f(x_{n-1})+f(x_n)\right].

The alternating $4,2,4,2,\ldots,4$ pattern is not optional bookkeeping; it comes from fitting quadratic pieces across pairs of subintervals.

Worked example

Simpson’s Rule is exact for cubics

For $f(x)=x^3$ on $[0,2]$ with $n=2$ , $\Delta x=1$ :

S_2=\frac13[f(0)+4f(1)+f(2)]=\frac13(0+4+8)=4.

The exact integral is also $[x^4/4]_0^2=4$ .

Interactive checku3a-simpson-01

Use Simpson’s Rule with $n=2$ to estimate $\int_0^2 x^3\,dx$ .

Your work stays on this device. No account or AI grader is used.

Show hint

Use values at 0, 1, and 2 with weights 1, 4, 1.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

After the explanation

Use the section idea

Reading lens

Make approximation error and limiting behavior explicit rather than hiding them behind a calculator result or an infinity symbol.

Mental model

Numerical rules replace a curve with simple local shapes; improper integrals replace a forbidden endpoint or infinite interval with a limit.

Decision

Choose the rule and partition, estimate scale and sign, or write the correct defining limit before evaluating.

Common trap

Treating an approximation as exact, using Simpson's Rule with an invalid partition, or substituting infinity as though it were a number.

Check yourself

Can you defend the estimate's scale or the improper integral's convergence from a written calculation?

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.