Intergration Calculator

Definite Integral Calculator (Simpson's Rule)

Use 'x' as the variable. For powers, use `x*x` or `Math.pow(x, 2)`. For trigonometric/exponential functions, use `Math.sin(x)`, `Math.cos(x)`, `Math.exp(x)`, `Math.log(x)`, etc.
A higher number of subintervals generally leads to greater accuracy.
Approximate Integral Value: Enter values and click 'Calculate'
function evaluateFunction(funcString, xValue) { var scope = { x: xValue }; for (var key in Math) { if (typeof Math[key] === 'function') { scope[key] = Math[key]; } else { scope[key] = Math[key]; // For constants like Math.PI, Math.E } } // Replace common power notation (x^2) with Math.pow(x, 2) for convenience // This regex handles 'x^2', 'var^3′, '10^x', etc. var processedFuncString = funcString.replace(/([a-zA-Z_][a-zA-Z0-9_]*|\d+(\.\d+)?)\^(\d+(\.\d+)?|x)/g, 'Math.pow($1, $3)'); try { // Use a function constructor to evaluate in a controlled scope var func = new Function('scope', 'with(scope) { return ' + processedFuncString + '; }'); return func(scope); } catch (e) { throw new Error("Invalid function expression: " + e.message); } } function calculateIntegral() { var functionInput = document.getElementById("functionInput").value; var lowerLimit = parseFloat(document.getElementById("lowerLimit").value); var upperLimit = parseFloat(document.getElementById("upperLimit").value); var numSubintervals = parseInt(document.getElementById("numSubintervals").value); var resultDisplay = document.getElementById("resultValue"); if (isNaN(lowerLimit) || isNaN(upperLimit) || isNaN(numSubintervals)) { resultDisplay.textContent = "Please enter valid numbers for all fields."; return; } if (lowerLimit >= upperLimit) { resultDisplay.textContent = "Lower limit must be less than upper limit."; return; } if (numSubintervals <= 0 || numSubintervals % 2 !== 0) { resultDisplay.textContent = "Number of subintervals (n) must be a positive, even integer."; return; } try { var h = (upperLimit – lowerLimit) / numSubintervals; var sum = evaluateFunction(functionInput, lowerLimit) + evaluateFunction(functionInput, upperLimit); for (var i = 1; i < numSubintervals; i++) { var x_i = lowerLimit + i * h; if (i % 2 === 1) { // Odd terms sum += 4 * evaluateFunction(functionInput, x_i); } else { // Even terms sum += 2 * evaluateFunction(functionInput, x_i); } } var integralValue = (h / 3) * sum; resultDisplay.textContent = integralValue.toFixed(8); // Display with 8 decimal places for precision } catch (e) { resultDisplay.textContent = "Error: " + e.message; } }

Understanding the Definite Integral Calculator

Integration is a fundamental concept in calculus, representing the accumulation of quantities and the area under a curve. While some functions can be integrated analytically (finding an exact antiderivative), many real-world functions or complex expressions do not have simple antiderivatives. This is where numerical integration comes into play.

What is Numerical Integration?

Numerical integration refers to a family of algorithms for calculating the numerical value of a definite integral. Instead of finding an exact symbolic solution, these methods approximate the area under the curve by dividing it into smaller, manageable shapes (like rectangles, trapezoids, or parabolas) and summing their areas.

Simpson's Rule Explained

This calculator uses Simpson's Rule, a popular and generally more accurate method for numerical integration compared to simpler methods like the Trapezoidal Rule or Riemann Sums. Simpson's Rule approximates the function with parabolic segments instead of straight lines or rectangles. This parabolic approximation allows it to capture the curvature of the function more effectively, leading to a more precise estimate of the integral.

The core idea is to divide the interval [a, b] into an even number of subintervals, 'n'. For each pair of subintervals, a parabola is fitted through three points on the function's curve. The area under these parabolas is then summed up to approximate the total integral.

The formula for Simpson's Rule is:

∫ab f(x) dx ≈ (h/3) * [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + … + 2f(xn-2) + 4f(xn-1) + f(xn)]

Where:

  • h = (b - a) / n is the width of each subinterval.
  • n is the number of subintervals, which must be an even number.
  • xi = a + i*h are the points along the interval.

How to Use This Calculator

  1. Function f(x): Enter the mathematical function you wish to integrate. Use 'x' as your variable. For powers, you can use `x*x` for x2, or `Math.pow(x, 2)`. For other mathematical functions, use JavaScript's `Math` object (e.g., `Math.sin(x)`, `Math.cos(x)`, `Math.exp(x)` for ex, `Math.log(x)` for natural logarithm, `Math.PI` for π).
  2. Lower Limit (a): Input the starting point of your integration interval.
  3. Upper Limit (b): Input the ending point of your integration interval.
  4. Number of Subintervals (n): Enter a positive, even integer. A larger 'n' generally leads to a more accurate approximation, but also requires more computation.
  5. Click "Calculate Integral" to see the approximate value.

Examples

  • Example 1: Integrate f(x) = x2 from 0 to 1.
    • Function f(x): x*x
    • Lower Limit (a): 0
    • Upper Limit (b): 1
    • Number of Subintervals (n): 100
    • Expected Result (analytical): 1/3 ≈ 0.33333333
  • Example 2: Integrate f(x) = sin(x) from 0 to π.
    • Function f(x): Math.sin(x)
    • Lower Limit (a): 0
    • Upper Limit (b): Math.PI
    • Number of Subintervals (n): 100
    • Expected Result (analytical): 2
  • Example 3: Integrate f(x) = ex from 0 to 1.
    • Function f(x): Math.exp(x)
    • Lower Limit (a): 0
    • Upper Limit (b): 1
    • Number of Subintervals (n): 100
    • Expected Result (analytical): e – 1 ≈ 1.71828183

Limitations

While powerful, numerical integration methods like Simpson's Rule have limitations:

  • The number of subintervals (n) must be an even integer.
  • The function must be continuous over the interval [a, b].
  • The accuracy of the approximation depends on the number of subintervals and the smoothness of the function. Highly oscillatory or discontinuous functions may require a very large 'n' for reasonable accuracy, or other specialized methods.
  • The calculator relies on JavaScript's `eval()`-like functionality for parsing the function string, which can be sensitive to syntax errors.

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