Midpoint Rule Calculator

Approximate \( \displaystyle \int_a^b f(x)\,dx \) using the composite midpoint rule. Enter \(f(x)\), \(a\), \(b\), and \(n\), then press Calculate.

Function \(f(x)\)

Use x as the variable. Allowed: + − * / ^, exp, ln/log, sin, cos, tan, sqrt, etc. Radians for trig.

Lower limit \(a\)

Start of the interval.

Upper limit \(b\)

End of the interval (must satisfy \(b>a\)).

Subintervals \(n\)

Positive integer. Step size \(h=(b-a)/n\).

Equation Preview

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Helping notes

\( \text{Composite midpoint rule: } \displaystyle \int_a^b f(x)\,dx \approx h\sum_{i=0}^{n-1} f\!\big(m_i\big),\; m_i=a+\big(i+\tfrac12\big)h,\; h=\tfrac{b-a}{n}. \) :contentReference[oaicite:1]{index=1}

\( \text{Required inputs: } f(x),\ a,\ b,\ n \text{ (no others needed).} \) :contentReference[oaicite:2]{index=2}

\( \text{Tip: Larger } n \text{ gives better accuracy; } h=(b-a)/n \text{ is the step width.} \) :contentReference[oaicite:3]{index=3}

Results

Approximation

Computation Details

Midpoints & Sample Values

What is a Midpoint Rule Calculator?

A Midpoint Rule Calculator is a numerical integration tool that estimates the area under a curve by sampling the integrand at the center of each subinterval. Instead of evaluating at endpoints (like the trapezoidal rule), the midpoint rule uses the average behavior around each subinterval’s center, which often provides better accuracy for smooth functions. The calculator computes the midpoint nodes, evaluates the function, and forms the weighted sum that approximates the definite integral \(\int_a^b f(x)\,dx\). It supports a single-panel formula and the composite midpoint rule with \(n\) equal subintervals, and it reports a theoretical error bound whenever a bound on the second derivative is known.

About the Midpoint Rule Calculator

The midpoint rule can be viewed as integrating a piecewise-constant approximation that matches \(f\) at each subinterval’s center. For sufficiently smooth \(f\), the global error scales like \(O(h^2)\), where \(h=(b-a)/n\) is the subinterval width. This makes the method second-order accurate, typically outperforming the left- and right-endpoint rectangle rules and rivaling the trapezoidal rule on many problems. The calculator constructs the node sequence \(x_i=a+ih\) and the midpoints \(m_i=\tfrac{x_{i-1}+x_i}{2}\), evaluates \(f(m_i)\), and accumulates the sum \(h\sum f(m_i)\). It also provides optional diagnostics: the list of midpoints, per-panel contributions, and a comparison with known antiderivatives for instructional examples.

How to Use this Midpoint Rule Calculator

Enter the integrand \(f(x)\) and the interval endpoints \(a<b\).
Choose the number of subintervals \(n\) (a positive integer). The step size is \(h=\tfrac{b-a}{n}\).
The calculator forms midpoints \(m_i=\tfrac{a+(i-1)h + a+ih}{2}=a+\big(i-\tfrac12\big)h\) and evaluates \(f(m_i)\).
It returns the midpoint estimate \(M_n=h\sum_{i=1}^{n} f(m_i)\) and, if available, an error bound using a supplied \(\max|f''|\).
Increase \(n\) until the estimate stabilizes to your desired precision.

Core Formulas (LaTeX)

Single-panel midpoint: \[ \int_a^b f(x)\,dx \;\approx\; (b-a)\,f\!\left(\frac{a+b}{2}\right). \]

Composite midpoint (equal spacing): \[ h=\frac{b-a}{n},\quad m_i=a+\left(i-\tfrac12\right)h,\quad M_n = h\sum_{i=1}^{n} f(m_i). \]

Error bound (smooth \(f\)): \[ \big|E_M\big|\;\le\;\frac{(b-a)}{24}\,h^{2}\,\max_{\xi\in[a,b]}\big|f''(\xi)\big|. \]

Examples (Illustrative)

Example 1 — Polynomial

\(f(x)=x^2\) on \([0,2]\), \(n=4\), \(h=0.5\). Midpoints: \(0.25,0.75,1.25,1.75\). Sum: \(f\)=\(0.0625+0.5625+1.5625+3.0625=5.25\). \(M_4=h\cdot\text{sum}=0.5\cdot5.25=2.625\). Exact integral: \(\tfrac{8}{3}\approx2.6667\).

Example 2 — Trigonometric

\(f(x)=\sin x\) on \([0,\pi]\), \(n=6\), \(h=\pi/6\). Midpoints: \(\pi/12,\;3\pi/12,\;\dots,\;11\pi/12\). \(M_6=\tfrac{\pi}{6}\sum \sin(\text{midpoints})\approx 1.989\) (exact integral is \(2\)).

Example 3 — Error estimate

\(f(x)=e^{x}\) on \([0,1]\). Here \(f''(x)=e^x\), so \(\max|f''|\le e\). With \(n=10\), \(h=0.1\): \(|E_M|\le \frac{1}{24}\,0.1^2\,e \approx 0.00113\).