Step by Step Integral Calculator

Step-by-Step Numerical Integral Calculator (Trapezoidal Rule)

Calculation Steps (Trapezoidal Rule):

h = (b - a) / n = (' + upperLimit + ' - ' + lowerLimit + ') / ' + numSubintervals + ' = ' + h.toFixed(4) + '

Integral ≈ (h/2) * [f(a) + 2f(x₁) + 2f(x₂) + ... + 2f(x_n-1) + f(b)]

S = f(a) + 2f(x₁) + ... + f(b)

'; var f_a = evaluateFunction(lowerLimit); if (isNaN(f_a)) return; // Stop if function evaluation failed sum += f_a; stepsHtml += '
• f(' + lowerLimit.toFixed(4) + ') = ' + f_a.toFixed(6) + '
'; for (var i = 1; i < numSubintervals; i++) { var x_i = lowerLimit + i * h; var f_x_i = evaluateFunction(x_i); if (isNaN(f_x_i)) return; // Stop if function evaluation failed sum += 2 * f_x_i; stepsHtml += '
• 2 * f(' + x_i.toFixed(4) + ') = 2 * ' + f_x_i.toFixed(6) + ' = ' + (2 * f_x_i).toFixed(6) + '
'; } var f_b = evaluateFunction(upperLimit); if (isNaN(f_b)) return; // Stop if function evaluation failed sum += f_b; stepsHtml += '
• f(' + upperLimit.toFixed(4) + ') = ' + f_b.toFixed(6) + '
'; stepsHtml += '

S = ' + sum.toFixed(6) + '

Integral ≈ (h/2) * S = (' + h.toFixed(4) + ' / 2) * ' + sum.toFixed(6) + ' = ' + integralApprox.toFixed(6) + '

Result:

f(x) = ' + functionStr + '

' + lowerLimit + '

' + upperLimit + '

Understanding Integration: A Step-by-Step Guide

Integration is a fundamental concept in calculus with wide-ranging applications in mathematics, physics, engineering, economics, and many other fields. At its core, integration can be thought of as the process of finding the area under a curve, the accumulation of quantities, or the inverse operation of differentiation.

What is an Integral?

An integral represents the sum of an infinite number of infinitesimally small quantities. There are two main types of integrals:

1. Indefinite Integral: Also known as an antiderivative, it represents a family of functions whose derivative is the given function. It includes an arbitrary constant 'C'. For example, the indefinite integral of 2x is x² + C.
2. Definite Integral: This calculates the exact numerical value of the area under a curve between two specified limits (a lower limit 'a' and an upper limit 'b'). It yields a single number, not a function.

While symbolic integration (finding indefinite integrals) can be complex and often requires advanced techniques, definite integrals can often be approximated using numerical methods, especially when a function doesn't have a simple antiderivative.

Numerical Integration: The Trapezoidal Rule

When it's difficult or impossible to find an exact antiderivative, or when you only have discrete data points, numerical integration methods come to the rescue. These methods approximate the definite integral by dividing the area under the curve into a series of simpler shapes, like rectangles or trapezoids, and summing their areas.

The Trapezoidal Rule is one of the simplest and most intuitive numerical integration techniques. Instead of approximating the area under the curve with rectangles (as in Riemann sums), it uses trapezoids. A trapezoid provides a better fit to the curve than a rectangle, generally leading to a more accurate approximation for a given number of subintervals.

How the Trapezoidal Rule Works:

Imagine you want to find the area under the curve of a function f(x) from x = a to x = b. The Trapezoidal Rule involves these steps:

1. Divide the Interval: The interval [a, b] is divided into n equal subintervals.
2. Calculate Width (h): The width of each subinterval, denoted as h, is calculated as h = (b - a) / n.
3. Form Trapezoids: Over each subinterval, a trapezoid is formed by connecting the points (xi, f(xi)) and (xi+1, f(xi+1)) with a straight line. The area of each trapezoid is (1/2) * (f(xi) + f(xi+1)) * h.
4. Sum the Areas: The total approximate integral is the sum of the areas of all these trapezoids.

The Trapezoidal Rule Formula:

The general formula for the Trapezoidal Rule is:

Integral ≈ (h/2) * [f(a) + 2f(x₁) + 2f(x₂) + ... + 2f(xn-1) + f(b)]

Where:

• h is the width of each subinterval.
• n is the number of subintervals.
• a is the lower limit of integration.
• b is the upper limit of integration.
• f(a) and f(b) are the function values at the endpoints.
• f(x₁), f(x₂), …, f(xn-1) are the function values at the interior points of the subintervals. Note that these interior points are multiplied by 2 in the sum.

How to Use the Step-by-Step Integral Calculator:

Our calculator uses the Trapezoidal Rule to approximate definite integrals. Follow these steps:

1. Enter the Function f(x): Type your mathematical function into the "Function f(x)" field. Use 'x' as the variable. For standard mathematical functions like sine, cosine, exponential, or logarithm, use JavaScript's Math object (e.g., Math.sin(x), Math.exp(x), Math.log(x), Math.pow(x, 2) for x squared).
2. Set Lower Limit (a): Enter the starting point of your integration interval.
3. Set Upper Limit (b): Enter the ending point of your integration interval.
4. Specify Number of Subintervals (n): Choose how many trapezoids you want to use for the approximation. A higher number of subintervals generally leads to a more accurate result but requires more computation.
5. Click "Calculate Integral": The calculator will then display the approximate integral value along with a step-by-step breakdown of how the Trapezoidal Rule was applied.

Example Calculation:

Let's approximate the definite integral of f(x) = x² from a = 0 to b = 2 using n = 4 subintervals.

1. Function: x*x
2. Lower Limit (a): 0
3. Upper Limit (b): 2
4. Number of Subintervals (n): 4

Steps the calculator performs:

• Calculate h: h = (2 - 0) / 4 = 0.5
• Identify x-values: x₀=0, x₁=0.5, x₂=1, x₃=1.5, x₄=2
• Evaluate f(x) at each point:
  • f(0) = 0² = 0
  • f(0.5) = 0.5² = 0.25
  • f(1) = 1² = 1
  • f(1.5) = 1.5² = 2.25
  • f(2) = 2² = 4
• Apply Trapezoidal Rule sum:

  Sum = f(0) + 2f(0.5) + 2f(1) + 2f(1.5) + f(2)

  Sum = 0 + 2(0.25) + 2(1) + 2(2.25) + 4

  Sum = 0 + 0.5 + 2 + 4.5 + 4 = 11

• Calculate Integral:

  Integral ≈ (h/2) * Sum = (0.5 / 2) * 11 = 0.25 * 11 = 2.75

The exact integral of x² from 0 to 2 is [x³/3] from 0 to 2, which is (2³/3) - (0³/3) = 8/3 ≈ 2.6667. Our approximation of 2.75 is quite close, and increasing n would bring it even closer.

Limitations:

This calculator provides a numerical approximation using the Trapezoidal Rule. It does not perform symbolic integration (finding antiderivatives). The accuracy of the approximation depends on the number of subintervals chosen; more subintervals generally lead to higher accuracy but also more computation. Also, be mindful of the function syntax; incorrect syntax will result in an error.