Definite Integral Calculator (Numerical Approximation)
This calculator approximates the definite integral of a function f(x) over an interval [a, b] using the Trapezoidal Rule. It provides a numerical approximation rather than a symbolic solution.
Approximation Steps (Trapezoidal Rule):
"; stepsHtml += "Function: f(x) =" + functionString + "";
stepsHtml += "Interval: [" + lowerLimit + ", " + upperLimit + "]";
stepsHtml += "Number of Subintervals (n): " + numSubintervals + "";
stepsHtml += "Width of each subinterval (h): (b – a) / n = (" + upperLimit + " – " + lowerLimit + ") / " + numSubintervals + " = " + h.toFixed(8) + "";
stepsHtml += "The Trapezoidal Rule formula is: Tn = (h/2) * [f(x0) + 2f(x1) + 2f(x2) + … + 2f(xn-1) + f(xn)]";
stepsHtml += "Calculating f(x) at each point (xi = a + i*h):";
stepsHtml += "- ";
// Calculate f(x_0)
var x0 = lowerLimit;
var fx0 = evaluateFunction(functionString, x0);
if (isNaN(fx0)) {
resultDiv.innerHTML = "Error: Could not evaluate function at x = " + x0 + ". Please check your function syntax.";
resultDiv.style.borderColor = '#dc3545';
resultDiv.style.backgroundColor = '#ffebeb';
resultDiv.style.color = '#dc3545';
return;
}
sumOfFValues += fx0;
stepsHtml += "
- f(x0) = f(
" + x0.toFixed(8) + ") =" + fx0.toFixed(8) + "";
// Calculate 2*f(x_i) for i = 1 to n-1
for (var i = 1; i < numSubintervals; i++) {
var xi = lowerLimit + i * h;
var fxi = evaluateFunction(functionString, xi);
if (isNaN(fxi)) {
resultDiv.innerHTML = "Error: Could not evaluate function at x = " + xi + ". Please check your function syntax.";
resultDiv.style.borderColor = '#dc3545';
resultDiv.style.backgroundColor = '#ffebeb';
resultDiv.style.color = '#dc3545';
return;
}
sumOfFValues += 2 * fxi;
if (numSubintervals <= 10 || i === 1 || i === numSubintervals – 1) { // Show detailed steps for small 'n' or first/last intermediate terms
stepsHtml += " - 2 * f(x" + i + ") = 2 * f(
" + xi.toFixed(8) + ") = 2 *" + fxi.toFixed(8) + "=" + (2 * fxi).toFixed(8) + "";
} else if (i === 2 && numSubintervals > 10) {
stepsHtml += " - … (intermediate terms omitted for brevity due to large 'n') … "; } } // Calculate f(x_n) var xn = upperLimit; var fxn = evaluateFunction(functionString, xn); if (isNaN(fxn)) { resultDiv.innerHTML = "Error: Could not evaluate function at x = " + xn + ". Please check your function syntax."; resultDiv.style.borderColor = '#dc3545'; resultDiv.backgroundColor = '#ffebeb'; resultDiv.style.color = '#dc3545'; return; } sumOfFValues += fxn; stepsHtml += "
- f(x" + numSubintervals + ") = f(
" + xn.toFixed(8) + ") =" + fxn.toFixed(8) + "";
stepsHtml += "
" + sumOfFValues.toFixed(8) + "";
stepsHtml += "Final Integral Approximation: (h/2) * Sum = (" + h.toFixed(8) + " / 2) * " + sumOfFValues.toFixed(8) + " = " + integral.toFixed(8) + "";
resultDiv.innerHTML = "Approximate Definite Integral: " + integral.toFixed(10) + "";
resultDiv.style.borderColor = '#28a745';
resultDiv.style.backgroundColor = '#e6ffe6';
resultDiv.style.color = '#1e7e34';
stepsDiv.innerHTML = stepsHtml;
}
Understanding the Definite Integral
The definite integral is a fundamental concept in calculus that represents the accumulation of quantities. Geometrically, for a non-negative function, it can be interpreted as the area of the region bounded by the function's graph, the x-axis, and two vertical lines (the lower and upper limits of integration). It's denoted as:
∫ab f(x) dx
Where:
f(x)is the function being integrated.ais the lower limit of integration.bis the upper limit of integration.dxindicates that the integration is with respect to the variablex.
Definite integrals have wide applications in physics (e.g., calculating work, displacement), engineering (e.g., fluid flow, stress analysis), economics (e.g., total cost, consumer surplus), and probability (e.g., probability density functions).
Why Numerical Approximation?
While many definite integrals can be solved analytically (finding an exact antiderivative and applying the Fundamental Theorem of Calculus), there are numerous functions for which finding an elementary antiderivative is impossible or extremely complex. In such cases, or when dealing with experimental data, numerical integration methods become indispensable. These methods approximate the value of the integral by dividing the area under the curve into many smaller, simpler shapes (like rectangles or trapezoids) and summing their areas.
The Trapezoidal Rule: A Step-by-Step Approach
This calculator uses the Trapezoidal Rule, a common and effective numerical integration technique. Instead of approximating the area under the curve with rectangles (as in Riemann sums), the Trapezoidal Rule uses trapezoids. A trapezoid generally provides a better fit to the curve than a rectangle, leading to a more accurate approximation for a given number of subintervals.
Here's how it works:
- Divide the Interval: The interval [a, b] is divided into
nequal subintervals. - Calculate Subinterval Width (h): The width of each subinterval, denoted as
h, is calculated ash = (b - a) / n. - Form Trapezoids: Over 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
(h/2) * [f(xi) + f(xi+1)]. - Sum the Areas: The total approximate integral is the sum of the areas of all these trapezoids. This leads to the formula:
Tn = (h/2) * [f(x0) + 2f(x1) + 2f(x2) + … + 2f(xn-1) + f(xn)]
Wherex0 = a,xn = b, andxi = a + i*hfori = 1, ..., n-1. Notice that the interior points are multiplied by 2 because they serve as an endpoint for two adjacent trapezoids.
The Role of 'n' (Number of Subintervals)
The parameter n, or the number of subintervals, directly impacts the accuracy of the approximation. Generally:
- Higher 'n': More subintervals mean narrower trapezoids, which fit the curve more closely. This results in a more accurate approximation of the definite integral.
- Lower 'n': Fewer subintervals lead to wider trapezoids and a less accurate approximation.
However, increasing 'n' also increases the computational effort. For most practical purposes, a sufficiently large 'n' (e.g., 100 to 1000 or more) provides a very good approximation.
Example Calculation: Integral of x2 from 0 to 1
Let's approximate the definite integral of f(x) = x2 from a=0 to b=1 using n=4 subintervals.
The exact value of this integral is [x3/3] from 0 to 1, which is 13/3 - 03/3 = 1/3 ≈ 0.33333333.
- Calculate h:
h = (1 - 0) / 4 = 0.25 - Identify xi points:
x0 = 0x1 = 0 + 1*0.25 = 0.25x2 = 0 + 2*0.25 = 0.50x3 = 0 + 3*0.25 = 0.75x4 = 1
- Evaluate f(x) at each point:
f(x0) = f(0) = 02 = 0f(x1) = f(0.25) = 0.252 = 0.0625f(x2) = f(0.50) = 0.502 = 0.25f(x3) = f(0.75) = 0.752 = 0.5625f(x4) = f(1) = 12 = 1
- Apply Trapezoidal Rule formula:
T4 = (h/2) * [f(x0) + 2f(x1) + 2f(x2) + 2f(x3) + f(x4)]T4 = (0.25/2) * [0 + 2(0.0625) + 2(0.25) + 2(0.5625) + 1]T4 = 0.125 * [0 + 0.125 + 0.5 + 1.125 + 1]T4 = 0.125 * [2.75]T4 = 0.34375
As you can see, 0.34375 is a good approximation of the exact value 0.33333333. If you increase n in the calculator (e.g., to 100 or 1000), the approximation will become even closer to the true value.