Integral Calculator with Steps

Numerical Integral Calculator (Trapezoidal Rule)

This calculator approximates the definite integral of the function f(x) = x² using the Trapezoidal Rule. The Trapezoidal Rule is a method for approximating the definite integral of a function by dividing the area under the curve into a series of trapezoids.

Lower Limit of Integration (a): Upper Limit of Integration (b): Number of Subintervals (n):

Understanding the Trapezoidal Rule Steps:

The Trapezoidal Rule approximates the area under a curve by dividing the interval [a, b] into 'n' subintervals of equal width, 'h'. For each subinterval, it forms a trapezoid and sums their areas.

The formula used is:

∫_a^b f(x) dx ≈ (h/2) * [f(a) + 2f(x₁) + 2f(x₂) + ... + 2f(x_n-1) + f(b)]

Where:

h = (b - a) / n (width of each subinterval)
a is the lower limit of integration
b is the upper limit of integration
n is the number of subintervals
f(x) is the function being integrated (in this calculator, f(x) = x²)
x_i = a + i * h are the points within the subintervals.

The calculator performs these steps internally to provide the approximate definite integral.

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What is an Integral?

In mathematics, an integral is a fundamental concept in calculus used to quantify the total accumulation of a quantity. It can be thought of as the continuous analog of a sum. The most common interpretation of a definite integral is the area of the region bounded by a function's graph, the x-axis, and two vertical lines (the limits of integration).

Types of Integrals:

Definite Integral: This type of integral has upper and lower limits, and its result is a single numerical value representing the area under the curve between those limits. For example, ∫_a^b f(x) dx.
Indefinite Integral (Antiderivative): This integral does not have limits and results in a family of functions whose derivative is the original function. It includes an arbitrary constant of integration (C). For example, ∫ f(x) dx = F(x) + C.

Why are Integrals Important?

Integrals have vast applications across various fields:

Physics: Calculating displacement from velocity, work done by a force, center of mass, moments of inertia.
Engineering: Determining volumes, surface areas, fluid flow, stress and strain analysis.
Economics: Calculating total cost or revenue from marginal cost/revenue functions, consumer and producer surplus.
Probability: Finding probabilities for continuous random variables (area under a probability density function).
Computer Graphics: Rendering realistic lighting and shadows.

Numerical Integration: Approximating the Area

While symbolic integration (finding an exact antiderivative) is ideal, it's not always possible for all functions. In such cases, or when dealing with empirical data, numerical integration methods are used to approximate the value of a definite integral. These methods divide the area under the curve into simpler geometric shapes (like rectangles or trapezoids) and sum their areas.

The Trapezoidal Rule Explained

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 approximation of the curve segment than a rectangle, especially for functions that are not constant over the subinterval.

Here's how it works:

Divide the Interval: The interval [a, b] over which you want to integrate is divided into 'n' equal subintervals.
Form Trapezoids: For each subinterval, a trapezoid is formed by connecting the function values at the endpoints of the subinterval with a straight line. The parallel sides of the trapezoid are the function values (heights), and the width of the trapezoid is the width of the subinterval.
Calculate Area: The area of a single trapezoid is given by (width / 2) * (sum of parallel sides). In our case, this is (h / 2) * [f(x_i) + f(x_i+1)].
Sum 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 are counted twice (once for each adjacent trapezoid), leading to the formula: (h/2) * [f(a) + 2f(x₁) + ... + 2f(x_n-1) + f(b)].

Example Calculation using the Calculator:

Let's say we want to approximate the definite integral of f(x) = x² from 0 to 1 using 100 subintervals.

Lower Limit (a): 0
Upper Limit (b): 1
Number of Subintervals (n): 100

The exact integral of x² from 0 to 1 is [x³/3] from 0 to 1, which equals (1³/3) – (0³/3) = 1/3 ≈ 0.333333.

Using the calculator with these inputs, you would get an approximate value very close to 0.333333, demonstrating the accuracy of the Trapezoidal Rule with a sufficient number of subintervals.

Increasing the number of subintervals (n) generally leads to a more accurate approximation, as the trapezoids fit the curve more closely.