Calculate the area under a curve using numerical integration (Simpson's Rule).
Use JavaScript syntax: Math.pow(x,2), Math.sin(x), Math.exp(x), Math.PI
Higher numbers increase accuracy (must be even).
Result
Understanding Definite Integrals
A definite integral represents the signed area between the graph of a function and the x-axis within a specific interval [a, b]. In calculus, this is the fundamental tool for calculating quantities that accumulate over time or space, such as distance, mass, or total work performed.
How This Calculator Works
This tool utilizes Simpson's 1/3 Rule, a numerical integration method that approximates the function using quadratic polynomials. The formula used is:
To find the area under the curve y = x² from x = 0 to x = 3:
Enter x * x in the Function field.
Set Lower Limit to 0.
Set Upper Limit to 3.
Click Calculate. The result will be 9.
function calculateIntegral() {
var funcInput = document.getElementById('mathFunction').value;
var a = parseFloat(document.getElementById('lowerLimit').value);
var b = parseFloat(document.getElementById('upperLimit').value);
var n = parseInt(document.getElementById('intervals').value);
var resultDiv = document.getElementById('integralResult');
var finalValue = document.getElementById('finalValue');
var formulaDisplay = document.getElementById('formulaDisplay');
var errorBox = document.getElementById('errorBox');
// Reset displays
errorBox.style.display = 'none';
resultDiv.style.display = 'none';
// Basic validation
if (!funcInput) {
showError("Please enter a function f(x).");
return;
}
if (isNaN(a) || isNaN(b) || isNaN(n)) {
showError("Please enter valid numerical limits and intervals.");
return;
}
if (n <= 0 || n % 2 !== 0) {
showError("Intervals (n) must be an even positive integer for Simpson's Rule.");
return;
}
try {
// Create a function evaluator
var f = function(x) {
// Replace common math shorthand for convenience if desired,
// but here we stick to standard JS Math object for safety.
return eval(funcInput);
};
// Test the function at x = a to see if it's valid
var testValue = f(a);
if (isNaN(testValue)) {
throw new Error("The function returned NaN at the lower limit. Check your syntax.");
}
// Simpson's Rule Logic
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 total = (h / 3) * sum;
// Display Result
finalValue.innerText = total.toLocaleString(undefined, {minimumFractionDigits: 0, maximumFractionDigits: 6});
formulaDisplay.innerText = "∫[" + a + " to " + b + "] (" + funcInput + ") dx";
resultDiv.style.display = 'block';
} catch (e) {
showError("Error in function expression: " + e.message);
}
}
function showError(msg) {
var errorBox = document.getElementById('errorBox');
errorBox.innerText = msg;
errorBox.style.display = 'block';
}