Calculus Calculator with Steps

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Trapezoidal Rule Numerical Integration Calculator

The Trapezoidal Rule is a numerical method for approximating the definite integral of a function. It works by dividing the area under the curve into a series of trapezoids instead of rectangles (as in Riemann sums). The sum of the areas of these trapezoids provides an approximation of the total area under the curve, and thus, the definite integral.

The formula for the Trapezoidal Rule is:

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

Where:

  • a is the lower limit of integration.
  • b is the upper limit of integration.
  • n is the number of subintervals (trapezoids).
  • h = (b - a) / n is the width of each subinterval.
  • xi = a + i*h are the points at which the function is evaluated.
f(x) = x^2 f(x) = x^3 f(x) = 1/x f(x) = sin(x) f(x) = e^x

How to Use the Trapezoidal Rule Calculator:

  1. Select Function: Choose the function f(x) you wish to integrate from the dropdown menu.
  2. Enter Limits: Input the lower limit (a) and the upper limit (b) of integration. Ensure b > a.
  3. Set Subintervals: Specify the number of subintervals (n) you want to use. A higher number of subintervals generally leads to a more accurate approximation but requires more calculation.
  4. Calculate: Click the "Calculate Integral" button.
  5. View Results: The calculator will display the approximate integral value and a detailed breakdown of the steps involved in applying the Trapezoidal Rule.

Example:

Let's approximate the integral of f(x) = x^2 from a = 0 to b = 2 using n = 4 subintervals.

  • Function: f(x) = x^2
  • Lower Limit (a): 0
  • Upper Limit (b): 2
  • Number of Subintervals (n): 4

Following the steps:

  1. h = (2 - 0) / 4 = 0.5
  2. The points are x0=0, x1=0.5, x2=1, x3=1.5, x4=2.
  3. The function values are:
    • f(0) = 0^2 = 0
    • f(0.5) = 0.5^2 = 0.25
    • f(1) = 1^2 = 1
    • f(1.5) = 1.5^2 = 2.25
    • f(2) = 2^2 = 4
  4. Applying the formula:
    Integral ≈ (0.5/2) * [f(0) + 2f(0.5) + 2f(1) + 2f(1.5) + f(2)]
    Integral ≈ 0.25 * [0 + 2(0.25) + 2(1) + 2(2.25) + 4]
    Integral ≈ 0.25 * [0 + 0.5 + 2 + 4.5 + 4]
    Integral ≈ 0.25 * [11]
    Integral ≈ 2.75

The exact integral of x^2 from 0 to 2 is [x^3/3] from 0 to 2, which is (2^3)/3 - (0^3)/3 = 8/3 ≈ 2.6667. The Trapezoidal Rule provides a good approximation.

function evaluateFunction(funcName, x) { switch (funcName) { case "x^2": return x * x; case "x^3": return x * x * x; case "1/x": // Handle division by zero for 1/x if (x === 0) { return NaN; // Or some other error indicator } return 1 / x; case "sin(x)": return Math.sin(x); case "e^x": return Math.exp(x); default: return 0; // Should not happen with dropdown } } function calculateTrapezoidalRule() { var funcName = document.getElementById("functionSelect").value; var lowerLimit = parseFloat(document.getElementById("lowerLimit").value); var upperLimit = parseFloat(document.getElementById("upperLimit").value); var numSubintervals = parseInt(document.getElementById("numSubintervals").value); var resultDiv = document.getElementById("trapezoidalResult"); var stepsDiv = document.getElementById("trapezoidalSteps"); resultDiv.innerHTML = ""; stepsDiv.innerHTML = ""; // Input validation if (isNaN(lowerLimit) || isNaN(upperLimit) || isNaN(numSubintervals) || lowerLimit >= upperLimit || numSubintervals <= 0) { resultDiv.innerHTML = "Please enter valid numbers for all fields. Upper limit must be greater than lower limit, and number of subintervals must be positive."; return; } var h = (upperLimit – lowerLimit) / numSubintervals; var sum = 0; var stepsHtml = "

Calculation Steps:

"; stepsHtml += "1. Calculate the width of each subinterval (h):"; stepsHtml += "h = (b – a) / n = (" + upperLimit + " – " + lowerLimit + ") / " + numSubintervals + " = " + h.toFixed(6) + ""; stepsHtml += "2. Evaluate the function at each point xi = a + i*h:"; stepsHtml += "
    "; var f_values = []; var hasInvalidFValue = false; for (var i = 0; i <= numSubintervals; i++) { var x_i = lowerLimit + i * h; var f_x_i = evaluateFunction(funcName, x_i); if (isNaN(f_x_i) || !isFinite(f_x_i)) { hasInvalidFValue = true; stepsHtml += "
  • x" + i + " = " + x_i.toFixed(6) + ", f(x" + i + ") = Undefined (e.g., division by zero or invalid domain)
    • "; } else { f_values.push(f_x_i); stepsHtml += "
  • x" + i + " = " + x_i.toFixed(6) + ", f(x" + i + ") = " + f_x_i.toFixed(6) + "
    • "; } } stepsHtml += "
"; if (hasInvalidFValue) { resultDiv.innerHTML = "Calculation stopped: Function is undefined or infinite at one or more points within the interval. Please check the function and limits."; stepsDiv.innerHTML = stepsHtml; // Show partial steps return; } stepsHtml += "3. Apply the Trapezoidal Rule formula: "; stepsHtml += "Integral ≈ (h/2) * [f(x0) + 2f(x1) + … + 2f(xn-1) + f(xn)]"; stepsHtml += "Sum = "; var sumTerms = []; for (var i = 0; i <= numSubintervals; i++) { var term; if (i === 0 || i === numSubintervals) { term = f_values[i]; sumTerms.push("f(x" + i + ") = " + term.toFixed(6)); } else { term = 2 * f_values[i]; sumTerms.push("2 * f(x" + i + ") = 2 * " + f_values[i].toFixed(6) + " = " + term.toFixed(6)); } sum += term; } stepsHtml += sumTerms.join(" + ") + ""; stepsHtml += "Total Sum = " + sum.toFixed(6) + ""; var integralApprox = (h / 2) * sum; stepsHtml += "4. Multiply by (h/2):"; stepsHtml += "Integral ≈ (" + h.toFixed(6) + " / 2) * " + sum.toFixed(6) + " = " + (h/2).toFixed(6) + " * " + sum.toFixed(6) + " = " + integralApprox.toFixed(6) + ""; resultDiv.innerHTML = "

Approximate Integral Value: " + integralApprox.toFixed(6) + "

"; stepsDiv.innerHTML = stepsHtml; }

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