Use this calculator to approximate the definite integral of a function over a given interval using the Trapezoidal Rule.
Use 'x' as the variable. For mathematical functions, use `Math.sin(x)`, `Math.cos(x)`, `Math.exp(x)`, `Math.log(x)`, `Math.sqrt(x)`, `Math.pow(x, y)`, `Math.PI`, `Math.E`.
Calculation Steps (Trapezoidal Rule):
.integration-calculator-container {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
background-color: #f9f9f9;
border: 1px solid #ddd;
padding: 20px;
border-radius: 8px;
max-width: 600px;
margin: 20px auto;
box-shadow: 0 2px 5px rgba(0,0,0,0.1);
}
.integration-calculator-container h2 {
color: #333;
text-align: center;
margin-bottom: 20px;
}
.integration-calculator-container p {
color: #555;
text-align: center;
margin-bottom: 25px;
}
.calculator-input-group {
margin-bottom: 15px;
}
.calculator-input-group label {
display: block;
margin-bottom: 5px;
font-weight: bold;
color: #444;
}
.calculator-input-group input[type="text"],
.calculator-input-group input[type="number"] {
width: calc(100% – 22px);
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
font-size: 16px;
}
.calculator-input-group small {
display: block;
margin-top: 5px;
color: #777;
font-size: 0.85em;
}
.integration-calculator-container button {
display: block;
width: 100%;
padding: 12px 20px;
background-color: #007bff;
color: white;
border: none;
border-radius: 4px;
font-size: 18px;
cursor: pointer;
transition: background-color 0.3s ease;
margin-top: 20px;
}
.integration-calculator-container button:hover {
background-color: #0056b3;
}
.calculator-result {
margin-top: 25px;
padding: 15px;
background-color: #e9f7ef;
border: 1px solid #d4edda;
border-radius: 4px;
font-size: 1.1em;
color: #155724;
text-align: center;
font-weight: bold;
}
.calculator-steps {
margin-top: 30px;
padding-top: 20px;
border-top: 1px dashed #ccc;
}
.calculator-steps h3 {
color: #333;
margin-bottom: 15px;
text-align: center;
}
.calculator-steps div {
background-color: #f0f8ff;
border: 1px solid #b8daff;
padding: 15px;
border-radius: 4px;
color: #004085;
line-height: 1.6;
}
.calculator-steps p {
margin-bottom: 8px;
text-align: left;
}
function calculateIntegration() {
var function_str = document.getElementById("function_str").value;
var lower_limit_str = document.getElementById("lower_limit").value;
var upper_limit_str = document.getElementById("upper_limit").value;
var num_intervals_str = document.getElementById("num_intervals").value;
var a = parseFloat(lower_limit_str);
var b = parseFloat(upper_limit_str);
var n = parseInt(num_intervals_str);
var resultDiv = document.getElementById("integration_result");
var stepsDiv = document.getElementById("integration_steps");
resultDiv.innerHTML = "";
stepsDiv.innerHTML = "";
if (isNaN(a) || isNaN(b) || isNaN(n) || !function_str) {
resultDiv.innerHTML = "Please enter valid numbers for all fields and a function.";
return;
}
if (n = b) {
resultDiv.innerHTML = "The upper limit (b) must be greater than the lower limit (a).";
return;
}
// Create a function from the user's string input
// This uses the Function constructor, which is safer than direct eval() for user-provided code.
var func;
try {
func = new Function('x', 'return ' + function_str + ';');
// Test the function with a dummy value to catch syntax errors early
func(0);
} catch (e) {
resultDiv.innerHTML = "Error in function syntax: " + e.message;
return;
}
var h = (b – a) / n;
var integral = 0;
var stepsHtml = "Function: f(x) = " + function_str + "";
stepsHtml += "Lower Limit (a): " + a + "";
stepsHtml += "Upper Limit (b): " + b + "";
stepsHtml += "Number of Subintervals (n): " + n + "";
stepsHtml += "Step Size (h): h = (b – a) / n = (" + b + " – " + a + ") / " + n + " = " + h.toFixed(6) + "";
stepsHtml += "The Trapezoidal Rule formula is: ";
stepsHtml += "Integral ≈ (h/2) * [f(x₀) + 2f(x₁) + 2f(x₂) + … + 2f(xₙ₋₁) + f(xₙ)]";
stepsHtml += "Where xᵢ = a + i * h";
stepsHtml += "Let's calculate the function values at each point:";
var sum_terms = [];
var f_a = func(a);
var f_b = func(b);
stepsHtml += "f(x₀) = f(" + a.toFixed(6) + ") = " + f_a.toFixed(6) + "";
sum_terms.push(f_a);
for (var i = 1; i < n; i++) {
var x_i = a + i * h;
var f_x_i = func(x_i);
integral += 2 * f_x_i;
stepsHtml += "2 * f(x" + i + ") = 2 * f(" + x_i.toFixed(6) + ") = 2 * " + f_x_i.toFixed(6) + " = " + (2 * f_x_i).toFixed(6) + "";
sum_terms.push(2 * f_x_i);
}
stepsHtml += "f(xₙ) = f(" + b.toFixed(6) + ") = " + f_b.toFixed(6) + "";
sum_terms.push(f_b);
integral = (f_a + integral + f_b) * (h / 2);
stepsHtml += "Sum of terms inside brackets: " + sum_terms.map(function(val){ return val.toFixed(6); }).join(' + ') + " = " + (f_a + (sum_terms.slice(1, -1).reduce(function(acc, val){ return acc + val; }, 0)) + f_b).toFixed(6) + "";
stepsHtml += "Integral ≈ (" + h.toFixed(6) + " / 2) * [" + (f_a + (sum_terms.slice(1, -1).reduce(function(acc, val){ return acc + val; }, 0)) + f_b).toFixed(6) + "]";
stepsHtml += "Integral ≈ " + integral.toFixed(8) + "";
resultDiv.innerHTML = "The approximate definite integral is: " + integral.toFixed(8) + "";
stepsDiv.innerHTML = stepsHtml;
}
Understanding Numerical Integration and the Trapezoidal Rule
Integration is a fundamental concept in calculus with wide-ranging applications, from calculating areas and volumes to determining total change from a rate of change. While symbolic integration provides exact solutions for many functions, not all functions can be integrated analytically. This is where numerical integration comes into play, offering powerful methods to approximate the value of a definite integral.
What is a Definite Integral?
A definite integral represents the net signed area between a function's graph and the x-axis over a specified interval [a, b]. If the function is above the x-axis, the area is positive; if below, it's negative. The notation for a definite integral is:
∫ab f(x) dx
Where 'f(x)' is the integrand, 'a' is the lower limit of integration, and 'b' is the upper limit of integration.
Why Numerical Integration?
Many real-world problems involve functions that are difficult or impossible to integrate symbolically. For instance, functions derived from experimental data or complex mathematical models often lack a simple antiderivative. Numerical integration techniques provide a way to estimate the value of these integrals to a desired degree of accuracy, which is crucial in fields like engineering, physics, economics, and computer graphics.
The Trapezoidal Rule Explained
The Trapezoidal Rule is one of the simplest and most intuitive methods for numerical integration. It approximates the area under the curve by dividing the integration interval [a, b] into a number of smaller subintervals. Over each subinterval, instead of approximating the area with rectangles (as in Riemann sums), it approximates the area with a trapezoid.
How it Works:
Divide the Interval: The interval [a, b] is divided into 'n' equal subintervals, each of width 'h'. The width 'h' is calculated as:
h = (b – a) / n
Form Trapezoids: For each subinterval [xᵢ, xᵢ₊₁], a trapezoid is formed by connecting the points (xᵢ, f(xᵢ)) and (xᵢ₊₁, f(xᵢ₊₁)) with a straight line. The area of a single trapezoid is given by:
Area of trapezoid = (1/2) * (base₁ + base₂) * height = (1/2) * (f(xᵢ) + f(xᵢ₊₁)) * h
Sum the Areas: The total approximate integral is the sum of the areas of all these trapezoids. When you sum them up, you'll notice that the interior function values (f(x₁), f(x₂), …, f(xₙ₋₁)) are counted twice (once for the left side of a trapezoid and once for the right side of the next), while the endpoints (f(x₀) and f(xₙ)) are counted once. This leads to the general formula for the Trapezoidal Rule:
The accuracy of the Trapezoidal Rule generally increases with the number of subintervals 'n'. As 'n' approaches infinity, the approximation approaches the true value of the integral. However, it's an approximation, and for functions with significant curvature, it might require a large 'n' to achieve high accuracy. Other numerical methods, like Simpson's Rule, can offer better accuracy for the same number of subintervals by approximating the curve with parabolas instead of straight lines.
Example Calculation:
Let's approximate the integral of f(x) = x² from a=0 to b=2 with n=4 subintervals.
Calculate h: h = (2 – 0) / 4 = 0.5
Determine x values: x₀=0, x₁=0.5, x₂=1, x₃=1.5, x₄=2
The exact integral of x² from 0 to 2 is [x³/3] from 0 to 2 = (2³/3) – (0³/3) = 8/3 ≈ 2.6667. Our approximation of 2.75 is reasonably close for only 4 subintervals.
This calculator provides a quick way to perform these numerical integrations, allowing you to experiment with different functions, limits, and numbers of subintervals to see how the approximation changes.