Integral Approximation Calculator
Approximate definite integrals numerically with the trapezoidal, midpoint, and Simpson rules. Enter any function, set bounds and intervals, and see live estimates with a graph.
What Is Integral Approximation?
Not every definite integral has a tidy antiderivative. Functions like $e^{-x^2}$, $\frac{\sin x}{x}$, or $\sqrt{1 + x^3}$ are famous for having no elementary closed form, and many practical integrals only exist as measured data points. Numerical integration solves this by slicing the area under a curve into simple shapes and adding them up. The finer the slices, the closer the total lands to the true integral.
This calculator approximates any definite integral $\int_a^b f(x)\,dx$ using three of the most widely taught quadrature rules: the trapezoidal rule, the midpoint rule, and Simpson's rule. Results update as you type, and you can compare all three methods side by side to see how they converge.
The Three Rules
All three methods start by dividing the interval $[a, b]$ into $n$ subintervals of equal width $h = \frac{b - a}{n}$ and evaluating $f$ at points inside each one.
Trapezoidal Rule
Each slice is treated as a trapezoid: a straight line connects the curve at the left and right edge of the slice, and the area is $\frac{h}{2}$ times the sum of the edge heights. Interior points are counted twice because each one is shared by two neighboring trapezoids.
$$T_n = \frac{h}{2}\left[f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n)\right]$$Midpoint Rule
Each slice is treated as a rectangle whose height is measured at the center of the slice. Midpoint rectangles overestimate and underestimate in a balanced way, which makes the rule noticeably more accurate than it looks.
$$M_n = h \sum_{i=1}^{n} f\left(x_{i-\frac{1}{2}}\right)$$Simpson's Rule
Pairs of adjacent slices are fitted with a parabola, and the weights follow the pattern 4, 2, 4, 2 through the interior points. Simpson's rule requires an even number of intervals; when you enter an odd $n$, this tool automatically uses the next even count for that method.
$$S_n = \frac{h}{3}\left[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 4f(x_{n-1}) + f(x_n)\right]$$How Accurate Is Each Rule?
The error of each method shrinks as the step width $h$ gets smaller, but at different rates:
- Trapezoidal rule: error is proportional to $h^2$, so halving the step width cuts the error by roughly a factor of four.
- Midpoint rule: also proportional to $h^2$, but with a smaller constant than the trapezoidal rule for typical smooth functions.
- Simpson's rule: proportional to $h^4$, so halving the step width cuts the error by roughly a factor of sixteen. For polynomials of degree three or lower, Simpson's rule is exact.
This is why Simpson's rule is usually the most accurate of the three, and why doubling the interval count is a cheap way to test whether your answer has stabilized.
A Worked Example
Approximating $\int_0^2 x^2\,dx = \frac{8}{3} \approx 2.6667$ with $n = 4$ slices gives a quick feel for the differences:
- Trapezoidal rule: $T_4 = 2.75$, which overshoots because straight lines cut above the convex parabola.
- Midpoint rule: $M_4 = 2.625$, which undershoots by about the same amount.
- Simpson's rule: $S_4 = 2.6667$, exact, because a quadratic is fitted perfectly by parabolic arcs.
Enter that function in the calculator, set the bounds to 0 and 2, and watch the same numbers appear in real time.
How to Use This Calculator
- Type your function of $x$ using standard notation such as
sin(x) + x^2 / 5,exp(-x^2), orsqrt(1 + x^3). Constants likepiandeare supported. - Set the lower bound $a$ and upper bound $b$ of the definite integral.
- Choose the number of intervals $n$. Larger values improve accuracy at the cost of more evaluations.
- Pick a method, or simply read the comparison table where all three appear at once.
- Expand the step-by-step breakdown to audit the interval width, the formula, and the substitution for the selected method.
The graph shades the area being measured, so you can sanity-check that the interval and function match what you intended before using the numeric result.
Practical Use Cases
- Probability and statistics: integrate normal, exponential, and other density functions that lack elementary antiderivatives.
- Physics and engineering: compute work, center of mass, fluid force, or charge from a function or set of readings.
- Calculus study: compare numerical results with exact answers to build intuition for how each quadrature rule behaves.
- Data analysis: approximate the area under an empirical curve when only sampled values are available.
Privacy
Everything is computed locally in your browser with JavaScript. The function, bounds, and results never leave your device, and the tool keeps working offline once the page has loaded.
Frequently Asked Questions
Which numerical integration method should I choose?
For most smooth functions, Simpson's rule gives the most accurate result for the same number of intervals, so it is a good default. Use the midpoint or trapezoidal rule when you want a simple estimate, when teaching the underlying concept, or when comparing convergence between methods.
How many intervals do I need for an accurate answer?
Start with 100 intervals, then double the count and see whether the result changes. If the value is stable to the number of digits you care about, the approximation has converged. Functions with sharp peaks or rapid oscillation need more intervals than smooth, gently varying functions.
Why does Simpson's rule use a different interval count sometimes?
Simpson's rule fits pairs of slices with parabolic arcs, which only works with an even number of intervals. If you enter an odd value such as 101, the calculator rounds up to 102 for Simpson's rule only and shows the count it used in the comparison table.
What function notation is supported?
You can use +, -, *, /, ^, parentheses, and functions such as sin, cos, tan, sqrt, abs, exp, log, and ln. Constants pi and e are also available. For example, exp(-x^2) or sqrt(1 + x^3) both work. If an expression cannot be parsed, the tool explains what to fix.
Why do I get an error about the function not being defined?
The rules only work when the function returns a finite value at every sampled point. Functions such as 1/x across x = 0, log(x) for x at or below 0, or sqrt(x) for negative x break the sampling. Adjust the bounds so the function is finite across the whole interval.
Is the result exact?
No. All three rules are approximations, and the true value differs slightly unless the function is a low-degree polynomial. Use the method comparison table and the step-by-step output to judge convergence, and increase the interval count when you need more precision.
Is my data sent to a server?
No. The function, bounds, and all computation run entirely in your browser. Nothing is uploaded, stored, or logged, and the tool works offline after the page loads.
Related Tools
Continue with these related math tools:
Related Tools
URL Extractor
TRIR Calculator
HTML to YAML Converter
Wedding Budget Calculator
Kilograms to Pounds Converter
TIMI STEMI Score Calculator
Tinetti Assessment Tool
Australian BSB Validator
Simpsons Diversity Index Calculator
Electric Field of a Point Charge Calculator
Metabolic Syndrome Calculator
Twin Dragon Curve Generator
Pool Calculator
Dice Roll Probability Calculator
Paracetamol Dosage Calculator
Cash Out Refinance Calculator
NDJSON Field Filter
Slenderness Ratio Calculator
GraphQL To Java Converter
Standard Notation Calculator