Step by Step Calculator Integral

Numerical Definite Integral Calculator (Trapezoidal Rule)

This calculator approximates the definite integral of a function f(x) over a given interval [a, b] using the Trapezoidal Rule. It provides a step-by-step breakdown of the approximation process.

function calculateIntegral() { var funcString = 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 resultDiv = document.getElementById("integralResult"); var stepsDiv = document.getElementById("stepByStepDetails"); resultDiv.innerHTML = ""; stepsDiv.innerHTML = ""; // Input validation if (isNaN(lowerLimit) || isNaN(upperLimit) || isNaN(numSubintervals) || numSubintervals <= 0) { resultDiv.innerHTML = "Please enter valid numerical inputs for limits and subintervals. Number of subintervals must be a positive integer."; return; } if (upperLimit <= lowerLimit) { resultDiv.innerHTML = "Upper limit must be greater than the lower limit."; return; } var func; try { // Create a function from the user's string input // Using new Function() is generally safer than eval() but still executes user-provided code. func = new Function('x', 'return ' + funcString + ';'); // Test the function with a dummy value to catch syntax errors early func(lowerLimit); } catch (e) { resultDiv.innerHTML = "Invalid function syntax. Please check your function f(x)."; stepsDiv.innerHTML = "Error details: " + e.message + ""; return; } var h = (upperLimit – lowerLimit) / numSubintervals; var sum = 0; var stepDetailsHtml = "

Trapezoidal Rule Steps:

"; stepDetailsHtml += "1. Calculate the width of each subinterval (h):"; stepDetailsHtml += "h = (b - a) / n = (" + upperLimit + " - " + lowerLimit + ") / " + numSubintervals + " = " + h.toFixed(6) + ""; stepDetailsHtml += "2. Evaluate the function f(x) at each subinterval endpoint:"; stepDetailsHtml += "

"; var fa = func(lowerLimit); sum += fa; stepDetailsHtml += "
• f(" + lowerLimit.toFixed(4) + ") = " + fa.toFixed(6) + " (first term)
"; var sumTermsForDisplay = []; sumTermsForDisplay.push(fa.toFixed(6)); for (var i = 1; i < numSubintervals; i++) { var x_i = lowerLimit + i * h; var fx_i = func(x_i); sum += 2 * fx_i; // Multiply by 2 for middle terms stepDetailsHtml += "
• f(" + x_i.toFixed(4) + ") = " + fx_i.toFixed(6) + " (multiplied by 2 in sum)
"; sumTermsForDisplay.push("2*" + fx_i.toFixed(6)); } var fb = func(upperLimit); sum += fb; stepDetailsHtml += "
• f(" + upperLimit.toFixed(4) + ") = " + fb.toFixed(6) + " (last term)
"; sumTermsForDisplay.push(fb.toFixed(6)); stepDetailsHtml += "

"; var integralApprox = (h / 2) * sum; stepDetailsHtml += "3. Apply the Trapezoidal Rule formula:"; stepDetailsHtml += "Integral ≈ (h / 2) * [f(a) + 2Σf(xi) + f(b)]"; stepDetailsHtml += "Substituting the values:"; stepDetailsHtml += "Integral ≈ (" + h.toFixed(6) + " / 2) * [" + sumTermsForDisplay.join(" + ") + "]"; stepDetailsHtml += "Integral ≈ " + (h / 2).toFixed(6) + " * [" + (sum).toFixed(6) + "]"; stepDetailsHtml += "Integral ≈ " + integralApprox.toFixed(6) + ""; resultDiv.innerHTML = "The approximate definite integral is: " + integralApprox.toFixed(6) + ""; stepsDiv.innerHTML = stepDetailsHtml; } .integral-calculator { font-family: Arial, sans-serif; max-width: 700px; margin: 20px auto; padding: 20px; border: 1px solid #ccc; border-radius: 8px; background-color: #f9f9f9; } .calculator-inputs label { display: block; margin-bottom: 5px; font-weight: bold; } .calculator-inputs input[type="text"], .calculator-inputs input[type="number"] { width: calc(100% – 22px); padding: 10px; margin-bottom: 15px; border: 1px solid #ddd; border-radius: 4px; } .calculator-inputs button { background-color: #007bff; color: white; padding: 10px 15px; border: none; border-radius: 4px; cursor: pointer; font-size: 16px; } .calculator-inputs button:hover { background-color: #0056b3; } .calculator-results { margin-top: 20px; padding-top: 15px; border-top: 1px solid #eee; } .calculator-results h3 { color: #333; margin-bottom: 10px; } .calculator-results p { margin-bottom: 8px; } .calculator-results ul { list-style-type: none; padding-left: 0; } .calculator-results li { margin-bottom: 5px; background-color: #eef; padding: 5px; border-radius: 3px; } .calculator-results code { background-color: #e0e0e0; padding: 2px 4px; border-radius: 3px; font-family: monospace; }

Understanding the Definite Integral and the Trapezoidal Rule

The definite integral is a fundamental concept in calculus used to find the accumulated quantity of a function over a specific interval. Geometrically, it represents the signed area under the curve of a function between two given points (the lower and upper limits of integration).

Symbolic vs. Numerical Integration

There are two main approaches to evaluating integrals:

1. Symbolic Integration: This involves finding an exact antiderivative of the function and then applying the Fundamental Theorem of Calculus. For example, the integral of x^2 is (1/3)x^3 + C. This method provides an exact solution but can be complex or impossible for many functions.
2. Numerical Integration: This method approximates the value of a definite integral using various numerical techniques. It's particularly useful when an antiderivative cannot be found analytically or when dealing with discrete data. This calculator uses a numerical method.

The Trapezoidal Rule Explained

The Trapezoidal Rule is a numerical method for approximating the definite integral. It works by dividing the area under the curve into a series of trapezoids instead of rectangles (as in Riemann sums). The area of each trapezoid is calculated, and then these areas are summed up to estimate the total area under the curve.

The formula for the Trapezoidal Rule is:

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

Where:

• a is the lower limit of integration.
• b is the upper limit of integration.
• n is the number of subintervals (trapezoids).
• h is the width of each subinterval, calculated as h = (b - a) / n.
• f(x) is the function being integrated.
• xi are the intermediate points within the interval.

As the number of subintervals (n) increases, the approximation generally becomes more accurate because the trapezoids fit the curve more closely.

How to Use This Calculator

1. Function f(x): Enter your function using JavaScript syntax. For example, x*x for x squared, Math.sin(x) for sine of x, Math.exp(x) for e to the power of x, or 2*x + 3 for a linear function. Remember that x is the variable.
2. Lower Limit (a): Enter the starting point of your integration interval.
3. Upper Limit (b): Enter the ending point of your integration interval.
4. Number of Subintervals (n): Specify how many trapezoids you want to use for the approximation. A higher number generally leads to a more accurate result but requires more computation.
5. Click "Calculate Integral" to see the approximate result and the step-by-step breakdown of the Trapezoidal Rule application.

Example Calculation

Let's approximate the definite integral of f(x) = x^2 from a = 0 to b = 1 with n = 4 subintervals.

Inputs:

• Function f(x): x*x
• Lower Limit (a): 0
• Upper Limit (b): 1
• Number of Subintervals (n): 4

Steps:

1. Calculate h: h = (1 - 0) / 4 = 0.25
2. Evaluate f(x) at endpoints:
   • f(0) = 0*0 = 0
   • f(0.25) = 0.25*0.25 = 0.0625
   • f(0.50) = 0.50*0.50 = 0.25
   • f(0.75) = 0.75*0.75 = 0.5625
   • f(1) = 1*1 = 1
3. Apply Trapezoidal Rule:

   Integral ≈ (0.25 / 2) * [f(0) + 2f(0.25) + 2f(0.50) + 2f(0.75) + f(1)]

   Integral ≈ 0.125 * [0 + 2*(0.0625) + 2*(0.25) + 2*(0.5625) + 1]

   Integral ≈ 0.125 * [0 + 0.125 + 0.5 + 1.125 + 1]

   Integral ≈ 0.125 * [2.75]

   Integral ≈ 0.34375

The exact integral of x^2 from 0 to 1 is [x^3 / 3] from 0 to 1, which is (1^3 / 3) - (0^3 / 3) = 1/3 ≈ 0.333333. As you can see, the numerical approximation is close, and would get closer with more subintervals.

Note on Function Input: This calculator uses JavaScript's new Function() constructor to interpret your function string. While generally safer than eval(), it still executes code provided by the user. Only use functions from trusted sources.