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Calculus I · Unit 3A · lesson

Midpoint and Trapezoidal Rules

Concept

Learning objectives

Compute numerical estimates from formulas or tables and reason from concavity about likely error direction.

Midpoint and Trapezoidal Rules

Explanation

Approximation is often the only honest option

Many definite integrals have no elementary antiderivative, and real data may be available only in a table. Numerical integration estimates the accumulated total directly from sampled values. The midpoint and trapezoidal rules both use local geometric approximations, but they model each subinterval differently: one uses a constant midpoint height, while the other uses a straight-line connection between endpoint values.

Error depends on interval width and curvature. A finer partition usually improves the estimate, but the direction of error can often be predicted from concavity. Trapezoids lie above a concave-up graph and below a concave-down graph; midpoint behavior is typically opposite. A complete numerical answer should state the rule, the partition, the units, and a sensible precision rather than presenting a long decimal as though a calculator had discovered exact truth.

The midpoint rule uses midpoint rectangles. The trapezoidal rule replaces each curve segment by a line segment:

Tn=Δx2[f(x0)+2f(x1)+⋯+2f(xn−1)+f(xn)].T_n=\frac{\Delta x}{2}\left[f(x_0)+2f(x_1)+\cdots+2f(x_{n-1})+f(x_n)\right].
Worked example

Trapezoidal estimate from data

Suppose

x0123f(x)12510\begin{array}{c|cccc} x&0&1&2&3\\ f(x)&1&2&5&10 \end{array}

With Δx=1\Delta x=1,

T3=12[1+2(2)+2(5)+10]=252.T_3=\frac12[1+2(2)+2(5)+10]=\frac{25}{2}.
Interactive checku3a-trap-01

Using one trapezoid, estimate ∫02x2 dx\int_0^2 x^2\,dx.

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Show hint

Average the endpoint heights 0 and 4, then multiply by width 2.

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Midpoint rectangles and trapezoids. Compare midpoint and trapezoidal approximations against a curved graph.
Read this graph as text

Midpoint rectangles and trapezoids. A smooth curve is approximated by midpoint rectangles or trapezoids. Compare midpoint and trapezoidal approximations against a curved graph. Do not claim one method always overestimates.

Every relationship in midpoint rectangles and trapezoids uses written labels together with distinct line styles, markers, or fill patterns; color is never the only carrier of meaning.

Why it matters: Compare midpoint and trapezoidal approximations against a curved graph.

Visual study

Midpoint rectangles and trapezoids. Compare midpoint and trapezoidal approximations against a curved graph.

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.

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