Calculas 2

Trapezoidal Rule Calculator for Definite Integrals

Use this calculator to approximate the definite integral of a quadratic function f(x) = Ax² + Bx + C over a given interval using the Trapezoidal Rule.

Function Coefficients (f(x) = Ax² + Bx + C)

Integration Interval and Subintervals

function calculateTrapezoidalRule() { var A = parseFloat(document.getElementById('coefficientA').value); var B = parseFloat(document.getElementById('coefficientB').value); var C = parseFloat(document.getElementById('constantC').value); var a = parseFloat(document.getElementById('lowerLimit').value); var b = parseFloat(document.getElementById('upperLimit').value); var n = parseInt(document.getElementById('numSubintervals').value); if (isNaN(A) || isNaN(B) || isNaN(C) || isNaN(a) || isNaN(b) || isNaN(n) || n <= 0) { document.getElementById('trapezoidalResult').innerHTML = 'Please enter valid numerical values for all fields, and ensure the number of subintervals is a positive integer.'; return; } // Define the function f(x) var f = function(x) { return A * x * x + B * x + C; }; var h = (b – a) / n; var sum = f(a) + f(b); // First and last terms for (var i = 1; i < n; i++) { var xi = a + i * h; sum += 2 * f(xi); // Middle terms are multiplied by 2 } var approximateIntegral = (h / 2) * sum; document.getElementById('trapezoidalResult').innerHTML = 'The approximate definite integral is: ' + approximateIntegral.toFixed(6) + ''; }

Understanding the Trapezoidal Rule in Calculus 2

In Calculus 2, one of the fundamental concepts is the definite integral, which represents the area under a curve. While analytical methods (antiderivatives) are often used, many functions do not have simple antiderivatives, or we might only have discrete data points. In such cases, numerical integration techniques become essential. The Trapezoidal Rule is one such method, providing an approximation of the definite integral.

What is the Trapezoidal Rule?

The Trapezoidal Rule approximates the area under a curve by dividing the region into a series of trapezoids instead of rectangles (as in Riemann sums). For each subinterval, it connects the function values at the endpoints with a straight line, forming a trapezoid. The area of each trapezoid is then calculated and summed to estimate the total area under the curve.

The Formula

Given a function f(x) to be integrated from x = a to x = b, and dividing the interval into n equal subintervals, each of width h = (b - a) / n, the Trapezoidal Rule formula is:

∫[a,b] f(x) dx ≈ (h / 2) * [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)]

Where:

• h is the width of each subinterval.
• x₀ = a, x₁ = a + h, …, xₙ = b are the endpoints of the subintervals.
• Notice that the first and last function values (f(x₀) and f(xₙ)) are multiplied by 1, while all intermediate function values are multiplied by 2. This is because the endpoints of the interval are part of only one trapezoid, while interior points are shared by two adjacent trapezoids.

Why Use the Trapezoidal Rule?

• Simplicity: It's relatively easy to understand and implement.
• Accuracy: Generally more accurate than basic Riemann sums (left, right, or midpoint) for the same number of subintervals because it uses a linear approximation between points, which often better fits the curve than a constant value.
• Practicality: Useful when dealing with experimental data where the function's analytical form might be unknown, and you only have discrete data points.

How to Use the Calculator

Our calculator focuses on quadratic functions of the form f(x) = Ax² + Bx + C. To use it:

1. Enter Coefficients: Input the values for A, B, and C that define your quadratic function. For example, for f(x) = x² + 2x + 1, you would enter A=1, B=2, C=1.
2. Set Limits: Enter the lower limit (a) and upper limit (b) of your integration interval.
3. Choose Subintervals: Specify the number of subintervals (n). A larger n generally leads to a more accurate approximation but requires more computation.
4. Calculate: Click the "Calculate Approximate Integral" button to see the result.

Example Calculation

Let's approximate the integral of f(x) = x² from x = 0 to x = 1 using n = 4 subintervals.

• Function: f(x) = 1x² + 0x + 0 (so A=1, B=0, C=0)
• Lower Limit (a): 0
• Upper Limit (b): 1
• Number of Subintervals (n): 4

Steps:

1. h = (1 - 0) / 4 = 0.25
2. The x-values are: x₀=0, x₁=0.25, x₂=0.5, x₃=0.75, x₄=1
3. Function values:
   • f(0) = 0² = 0
   • f(0.25) = 0.25² = 0.0625
   • f(0.5) = 0.5² = 0.25
   • f(0.75) = 0.75² = 0.5625
   • f(1) = 1² = 1
4. Apply the formula: (0.25 / 2) * [f(0) + 2f(0.25) + 2f(0.5) + 2f(0.75) + f(1)]
0.125 * [0 + 2(0.0625) + 2(0.25) + 2(0.5625) + 1]
0.125 * [0 + 0.125 + 0.5 + 1.125 + 1]
0.125 * [2.75] = 0.34375

The calculator would output approximately 0.343750. (The exact integral of x² from 0 to 1 is 1/3 or 0.333333…). As you increase 'n', the approximation gets closer to the exact value.