Calculate the definite integral of a function using numerical approximation (Trapezoidal Rule).
Approximate Area Under Curve
Understanding Area Under the Curve (Definite Integration)
The area under the curve, in calculus, refers to the area between a function's graph and the x-axis over a specified interval. This concept is fundamental to understanding definite integration. A definite integral, denoted as $\int_{a}^{b} f(x) \, dx$, represents the net signed area between the function $f(x)$ and the x-axis from $x=a$ (the lower limit) to $x=b$ (the upper limit).
Why Calculate Area Under the Curve?
Calculating the area under a curve has numerous applications across various fields:
Physics: Calculating displacement from velocity-time graphs, work done from force-distance graphs.
Engineering: Analyzing signal processing, fluid dynamics, and stress/strain relationships.
Economics: Determining total cost, revenue, or profit over time.
Probability and Statistics: Finding probabilities from probability density functions (PDFs).
Biology: Modeling population growth or decay.
Methods for Calculating Area Under the Curve
While analytical methods (finding the antiderivative) are precise, they are not always feasible for complex functions. Numerical integration methods provide approximations. This calculator uses the Trapezoidal Rule, a common and effective numerical technique.
The Trapezoidal Rule Explained
The Trapezoidal Rule approximates the area under the curve by dividing the interval $[a, b]$ into $n$ smaller subintervals of equal width, $\Delta x = \frac{b-a}{n}$. In each subinterval, the curve is approximated by a straight line, forming a trapezoid. The area of each trapezoid is calculated, and these areas are summed up to approximate the total area under the curve.
The exact integral of $x^2$ from 0 to 2 is $\frac{x^3}{3} \Big|_0^2 = \frac{2^3}{3} – \frac{0^3}{3} = \frac{8}{3} \approx 2.667$. The approximation is close, and accuracy increases with more intervals.
function calculateArea() {
var functionString = document.getElementById("functionInput").value;
var a = parseFloat(document.getElementById("lowerBound").value);
var b = parseFloat(document.getElementById("upperBound").value);
var n = parseInt(document.getElementById("numIntervals").value);
var resultDiv = document.getElementById("result");
var resultValueDiv = document.getElementById("result-value");
// Clear previous results
resultDiv.style.display = 'none';
resultValueDiv.textContent = ";
// Input validation
if (isNaN(a) || isNaN(b) || isNaN(n) || n = b) {
alert("Lower bound (a) must be less than the upper bound (b).");
return;
}
// Define a safe evaluation function
function safeEval(funcStr, xVal) {
try {
// Replace common math functions and constants
funcStr = funcStr.replace(/sin/g, 'Math.sin');
funcStr = funcStr.replace(/cos/g, 'Math.cos');
funcStr = funcStr.replace(/tan/g, 'Math.tan');
funcStr = funcStr.replace(/sqrt/g, 'Math.sqrt');
funcStr = funcStr.replace(/log/g, 'Math.log'); // Natural log
funcStr = funcStr.replace(/log10/g, 'Math.log10');
funcStr = funcStr.replace(/exp/g, 'Math.exp');
funcStr = funcStr.replace(/pi/g, 'Math.PI');
funcStr = funcStr.replace(/e/g, 'Math.E');
// Handle exponentiation (e.g., x^2)
funcStr = funcStr.replace(/\^/g, '**');
// Use a limited scope for evaluation
var scope = { x: xVal };
with(scope) {
var result = eval(funcStr);
if (typeof result !== 'number' || !isFinite(result)) {
throw new Error("Invalid function result");
}
return result;
}
} catch (e) {
console.error("Error evaluating function:", e);
throw new Error("Could not evaluate function '" + funcStr + "' at x = " + xVal + ". Check syntax and available functions (sin, cos, tan, sqrt, log, log10, exp, pi, e).");
}
}
var deltaX = (b – a) / n;
var sum = 0;
try {
// Add the first term f(x0)
sum += safeEval(functionString, a);
// Add the middle terms 2*f(xi)
for (var i = 1; i < n; i++) {
var x_i = a + i * deltaX;
sum += 2 * safeEval(functionString, x_i);
}
// Add the last term f(xn)
sum += safeEval(functionString, b);
var area = (deltaX / 2) * sum;
resultValueDiv.textContent = area.toFixed(6); // Display with 6 decimal places
resultDiv.style.display = 'block';
} catch (e) {
alert("Calculation Error: " + e.message);
}
}