Use standard math notation: ^ for power, * for multiply, / for divide. (e.g., Math.sin(x), Math.exp(x))
Higher values provide better accuracy (max 100,000).
Integration Result:
function calculateIntegral() {
var funcInput = document.getElementById('calcFunction').value;
var a = parseFloat(document.getElementById('lowerBound').value);
var b = parseFloat(document.getElementById('upperBound').value);
var n = parseInt(document.getElementById('intervals').value);
var resultDiv = document.getElementById('calcResult');
var resultText = document.getElementById('resultText');
if (!funcInput || isNaN(a) || isNaN(b) || isNaN(n)) {
alert("Please fill in all fields with valid numbers.");
return;
}
if (n < 2) n = 2;
if (n % 2 !== 0) n++; // Ensure n is even for Simpson's Rule
try {
// Pre-process function string
var preparedFunc = funcInput.replace(/\^/g, "**");
// Allow common math functions without Math. prefix
var mathSymbols = ["sin", "cos", "tan", "exp", "log", "sqrt", "PI", "pow", "abs"];
mathSymbols.forEach(function(sym) {
var re = new RegExp("\\b" + sym + "\\b", "g");
preparedFunc = preparedFunc.replace(re, "Math." + sym);
});
// Create a function object
var f = new Function("x", "return " + preparedFunc);
// Numerical Integration using Simpson's 1/3 Rule
var h = (b – a) / n;
var sum = f(a) + f(b);
for (var i = 1; i < n; i++) {
var x = a + i * h;
if (i % 2 === 0) {
sum += 2 * f(x);
} else {
sum += 4 * f(x);
}
}
var finalIntegral = (h / 3) * sum;
resultText.innerHTML = "∫ ≈ " + finalIntegral.toLocaleString(undefined, {minimumFractionDigits: 6, maximumFractionDigits: 6});
document.getElementById('methodUsed').innerText = "Calculated using Simpson's Rule with " + n + " intervals.";
resultDiv.style.display = 'block';
} catch (error) {
alert("Error in function syntax. Please check your math expression.");
console.error(error);
}
}
Understanding Definite Integration
Integration is a fundamental concept in calculus, representing the accumulation of quantities and the area under a curve. While indefinite integration focuses on finding the antiderivative, definite integration calculates the net signed area between a function and the x-axis within a specific interval [a, b].
How This Calculator Works
Our tool uses Simpson's 1/3 Rule, a method for numerical integration that provides high accuracy by approximating the function with quadratic polynomials. Unlike simpler methods like the Trapezoidal rule, Simpson's rule accounts for the curvature of the function, making it ideal for smooth curves.
The Formula:
∫ab f(x) dx ≈ (h/3) * [f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + … + f(xₙ)]
Where h = (b – a) / n and n is an even number of sub-intervals.
Practical Examples
Linear Function: Integrating x from 0 to 2 yields an area of 2.
Parabola: Integrating x^2 from 0 to 3 yields an area of 9.
Trigonometry: Integrating sin(x) from 0 to Math.PI yields exactly 2.
Exponential Growth: Integrating exp(x) from 0 to 1 yields approximately 1.718.
Common Syntax Rules
To ensure the calculator processes your math correctly, follow these conventions:
Operation
Input Style
Example
Multiplication
*
3*x
Exponentiation
^ or **
x^2
Square Root
sqrt(x)
sqrt(x + 5)
Sine / Cosine
sin(x) / cos(x)
sin(x) * 2
Why Use Numerical Integration?
Many real-world functions do not have an elementary antiderivative (the "Gaussian Integral" or exp(-x^2) is a famous example). In engineering, physics, and data science, we often need the area under a curve where finding the exact formula is impossible. Numerical integration provides a fast, reliable approximation that is accurate enough for most scientific applications.