Numerical Integration Calculator (Trapezoidal Rule)
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. The calculator will also display the step-by-step process.
Approximate Integral:
Understanding Numerical Integration and the Trapezoidal Rule
Integration is a fundamental concept in calculus used to find the total accumulation of a quantity. Geometrically, a definite integral represents the area under the curve of a function between two specified limits. While analytical integration provides exact solutions for many functions, some functions are difficult or impossible to integrate analytically. In such cases, or when dealing with discrete data, numerical integration methods are employed to approximate the integral.
What is the Trapezoidal Rule?
The Trapezoidal Rule is one of the simplest and most commonly used numerical integration techniques. It approximates the area under a curve by dividing the region 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 definite integral.
The Formula
For a function f(x) integrated from a to b with n subintervals, the Trapezoidal Rule formula is:
∫ab f(x) dx ≈ (h/2) * [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xn-1) + f(xn)]
Where:
h = (b - a) / nis the width of each subinterval.x₀ = a(the lower limit).xn = b(the upper limit).xᵢ = a + i * hfori = 1, 2, ..., n-1are the intermediate points.
Essentially, it averages the left and right Riemann sums. Each trapezoid's area is calculated as (base1 + base2) / 2 * height, where the bases are the function values at the endpoints of the subinterval, and the height is h.
How to Use the Calculator
- Enter Coefficients: Input the coefficients
a,b, andcfor your quadratic functionf(x) = ax² + bx + c. For example, forf(x) = x², you would entera=1, b=0, c=0. - Set Limits: Enter the
Lower Limit (a)andUpper Limit (b)of integration. - Choose Subintervals: Specify the
Number of Subintervals (n). A higher number of subintervals generally leads to a more accurate approximation, but also more calculations. - Calculate: Click the "Calculate Integral" button to see the approximate integral value and the detailed steps involved in the Trapezoidal Rule calculation.
Example Calculation
Let's approximate the integral of f(x) = x² from 0 to 1 using 4 subintervals.
a = 1, b = 0, c = 0(forf(x) = x²)- Lower Limit (a) = 0
- Upper Limit (b) = 1
- Number of Subintervals (n) = 4
Steps:
- Calculate
h = (b - a) / n = (1 - 0) / 4 = 0.25. - The points are
x₀=0, x₁=0.25, x₂=0.5, x₃=0.75, x₄=1. - Calculate function values:
f(0) = 0² = 0f(0.25) = 0.25² = 0.0625f(0.5) = 0.5² = 0.25f(0.75) = 0.75² = 0.5625f(1) = 1² = 1
- Apply the formula:
Integral ≈ (0.25 / 2) * [f(0) + 2f(0.25) + 2f(0.5) + 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 x² from 0 to 1 is [x³/3] from 0 to 1, which is 1/3 ≈ 0.33333. As you can see, the Trapezoidal Rule provides a good approximation.
Calculation Steps:
"; stepsHTML += "Function: f(x) = " + coeffA + "x² + " + coeffB + "x + " + coeffC + ""; stepsHTML += "Lower Limit (a): " + lowerLimit + ""; stepsHTML += "Upper Limit (b): " + upperLimit + ""; stepsHTML += "Number of Subintervals (n): " + numSubintervals + ""; stepsHTML += "Width of each subinterval (h) = (b – a) / n = (" + upperLimit + " – " + lowerLimit + ") / " + numSubintervals + " = " + h.toFixed(6) + ""; stepsHTML += "Applying the Trapezoidal Rule formula:"; stepsHTML += "Integral ≈ (h/2) * [f(x₀) + 2f(x₁) + ... + 2f(xn-1) + f(xn)]";
stepsHTML += "Where x₀ = a, xn = b, and xᵢ = a + i*h";
stepsHTML += "Calculating f(x) for each point:- ";
var intermediateSumTerms = [];
for (var i = 1; i < numSubintervals; i++) {
var x_i = lowerLimit + i * h;
var f_x_i = f(x_i);
sum += 2 * f_x_i;
intermediateSumTerms.push("2 * f(" + x_i.toFixed(6) + ") = 2 * " + f_x_i.toFixed(6) + " = " + (2 * f_x_i).toFixed(6));
stepsHTML += "
- f(" + x_i.toFixed(6) + ") = " + f_x_i.toFixed(6) + " (multiplied by 2 in sum) "; } stepsHTML += "
- f(" + lowerLimit.toFixed(6) + ") = " + f(lowerLimit).toFixed(6) + " "; stepsHTML += "
- f(" + upperLimit.toFixed(6) + ") = " + f(upperLimit).toFixed(6) + " "; stepsHTML += "