An integral, in calculus, is a fundamental concept that represents the area under a curve. It's essentially the reverse operation of differentiation. Integrals can be broadly categorized into two types:
Indefinite Integral: This finds the antiderivative of a function, which is a family of functions whose derivative is the original function. It's denoted as ∫ f(x) dx.
Definite Integral: This calculates the net area between a function's curve and the x-axis over a specified interval [a, b]. It's denoted as ∫ab f(x) dx.
The definite integral is what this calculator primarily helps approximate. It answers questions like:
What is the total distance traveled given a velocity function over a time interval?
What is the volume of a solid of revolution?
What is the work done by a variable force?
What is the accumulated change of a quantity whose rate of change is known?
Numerical Integration (Approximation)
While analytical solutions (finding the exact antiderivative) are preferred when possible, many functions do not have simple antiderivatives or the antiderivative is too complex to find. In such cases, numerical methods are used to approximate the definite integral. This calculator uses a common numerical method, the Trapezoidal Rule (or a similar Riemann sum variant depending on the exact implementation complexity) to approximate the area under the curve.
The method works by dividing the interval [a, b] into a large number n of smaller subintervals. In each subinterval, the area under the curve is approximated by a simple geometric shape, such as a rectangle (Riemann Sum) or a trapezoid (Trapezoidal Rule). Summing the areas of these shapes provides an approximation of the total area. Increasing the number of steps (n) generally leads to a more accurate result, but also requires more computation.
How to Use This Calculator
1. Function f(x): Enter the mathematical function you want to integrate. You can use standard mathematical notation (e.g., x^2 for x squared, sin(x), cos(x), exp(x) for e^x, log(x) for natural logarithm). For complex functions, parentheses are important.
2. Lower Limit (a): Enter the starting value of your integration interval.
3. Upper Limit (b): Enter the ending value of your integration interval.
4. Number of Steps (n): This determines the precision of the approximation. A higher number yields a more accurate result but takes longer to compute.
5. Click "Calculate Integral". The result displayed is the approximate value of the definite integral ∫ab f(x) dx.
Note: This calculator uses a simplified numerical method for approximation and may not handle all complex functions or edge cases perfectly. For highly accurate or symbolic integration, dedicated mathematical software is recommended.
function calculateIntegral() {
var funcString = document.getElementById("function").value.trim();
var a = parseFloat(document.getElementById("lowerBound").value);
var b = parseFloat(document.getElementById("upperBound").value);
var n = parseInt(document.getElementById("steps").value);
var resultDiv = document.getElementById("result");
// Clear previous error messages or results
resultDiv.innerText = "";
resultDiv.style.backgroundColor = "var(–success-green)";
// — Input Validation —
if (!funcString) {
resultDiv.innerText = "Error: Function cannot be empty.";
resultDiv.style.backgroundColor = "#dc3545"; // Red for error
return;
}
if (isNaN(a)) {
resultDiv.innerText = "Error: Invalid Lower Limit (a).";
resultDiv.style.backgroundColor = "#dc3545";
return;
}
if (isNaN(b)) {
resultDiv.innerText = "Error: Invalid Upper Limit (b).";
resultDiv.style.backgroundColor = "#dc3545";
return;
}
if (isNaN(n) || n b) {
// Swap limits and negate the result
var temp = a;
a = b;
b = temp;
// We'll multiply the final result by -1
}
// — Function Evaluation Helper —
// This is a simplified math parser. For a robust solution,
// a dedicated library (like math.js) would be used.
// This handles basic arithmetic, common functions, and exponentiation.
function evaluateFunction(fStr, xVal) {
try {
// Replace common notations
var processedFunc = fStr.toLowerCase()
.replace(/sin/g, 'Math.sin')
.replace(/cos/g, 'Math.cos')
.replace(/tan/g, 'Math.tan')
.replace(/log/g, 'Math.log') // Natural log
.replace(/ln/g, 'Math.log') // Natural log alias
.replace(/exp/g, 'Math.exp')
.replace(/sqrt/g, 'Math.sqrt')
.replace(/pi/g, 'Math.PI');
// Handle exponentiation (e.g., x^2)
// This regex looks for patterns like `number^number`, `x^number`, `number^x`, `x^x`
// It's important to handle operator precedence, so we iteratively apply it.
// A more robust parser would use an Abstract Syntax Tree (AST).
// For simplicity here, we'll rely on JS's Math.pow after some preprocessing.
// Let's try to make `^` work by replacing it with `**` if the browser supports it or `Math.pow`.
// Since we are using `var`, let's assume older JS compatibility might be an issue with `**`.
// We will replace `^` with `Math.pow(base, exponent)`.
// This requires careful parsing. A simpler approach for this context:
// Replace `x^n` patterns.
processedFunc = processedFunc.replace(/x\^(\d+(\.\d+)?)/g, 'Math.pow(x,$1)');
processedFunc = processedFunc.replace(/(\d+(\.\d+)?)\^x/g, 'Math.pow($1,x)');
processedFunc = processedFunc.replace(/x\^x/g, 'Math.pow(x,x)');
processedFunc = processedFunc.replace(/(\d+(\.\d+)?)\^(\d+(\.\d+)?)/g, 'Math.pow($1,$3)');
// Define the variable 'x' in the scope for evaluation
var x = xVal;
// Evaluate the function string
// Using Function constructor is generally safer than eval for untrusted input,
// but still carries risks if the input is malicious.
// For this calculator context, it's a common approach for dynamic function evaluation.
var result = new Function('x', 'Math', 'processedFunc', 'return ' + processedFunc)(x, Math, processedFunc);
if (typeof result !== 'number' || isNaN(result)) {
console.error("Evaluation resulted in NaN or non-number for:", processedFunc, "at x =", xVal);
return NaN; // Indicate failure
}
return result;
} catch (e) {
console.error("Error evaluating function:", e, "with input string:", fStr);
return NaN; // Indicate failure
}
}
// — Numerical Integration (Trapezoidal Rule) —
var h = (b – a) / n; // Width of each subinterval
var integralSum = 0;
var i = 0;
for (i = 0; i document.getElementById("upperBound").value) {
integralSum = -integralSum;
}
resultDiv.innerText = "Integral Result: " + integralSum.toFixed(6); // Display with 6 decimal places
}