Calculating Integrals – Loan calculator

Definite Integral Calculator (Trapezoidal Rule)

This calculator approximates the definite integral of a function over a given interval using the Trapezoidal Rule. The Trapezoidal Rule is a numerical method for approximating the definite integral, which is the area under the curve of a function between two points.

Function f(x): Use 'x' as the variable. For powers, use `Math.pow(x, y)`. For trigonometric functions, use `Math.sin(x)`, `Math.cos(x)`, etc. For e^x, use `Math.exp(x)`.

Lower Limit (a):

Upper Limit (b):

Number of Intervals (n):

Result:

Understanding Integrals and the Trapezoidal Rule

In mathematics, an integral is a fundamental concept in calculus used to find the total accumulation of a quantity. It can be thought of as the area under the curve of a function on a graph. When we talk about a definite integral, we are calculating this area between two specific points (a lower limit and an upper limit) on the x-axis.

Why Numerical Integration?

While many integrals can be solved analytically (using symbolic methods), some functions are too complex or even impossible to integrate exactly. In such cases, or when dealing with experimental data, numerical integration methods come to the rescue. These methods approximate the value of the definite integral by dividing the area under the curve into many smaller, simpler shapes whose areas can be easily calculated and summed up.

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. Here's how it works:

The interval [a, b] (from the lower limit to the upper limit) is divided into 'n' equal sub-intervals.
For each sub-interval, a trapezoid is formed by connecting the function's value at the start of the sub-interval to its value at the end of the sub-interval with a straight line.
The area of each trapezoid is calculated. The formula for the area of a trapezoid is (base1 + base2) / 2 * height. In our case, the 'bases' are the function values (y-coordinates) at the ends of the sub-interval, and the 'height' is the width of the sub-interval (h).
All these individual trapezoidal areas are summed up to get the total approximate area under the curve, which is the definite integral.

The formula for the Trapezoidal Rule 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 is the width of each sub-interval.
f(a) and f(b) are the function values at the lower and upper limits.
f(x_i) are the function values at the intermediate points within the interval.

The accuracy of the approximation generally increases as the number of intervals (n) increases, as the trapezoids fit the curve more closely.

Example Calculation:

Let's approximate the integral of f(x) = x² from x = 0 to x = 1 using n = 4 intervals.

Function: f(x) = x*x
Lower Limit (a): 0
Upper Limit (b): 1
Number of Intervals (n): 4

First, calculate h:

h = (b - a) / n = (1 - 0) / 4 = 0.25

The x-values for the trapezoids are: 0, 0.25, 0.5, 0.75, 1.

Now, evaluate f(x) at these points:

f(0) = 0² = 0
f(0.25) = 0.25² = 0.0625
f(0.5) = 0.5² = 0.25
f(0.75) = 0.75² = 0.5625
f(1) = 1² = 1

Apply the Trapezoidal Rule 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 1³/3 - 0³/3 = 1/3 ≈ 0.333333. As you can see, with just 4 intervals, the Trapezoidal Rule provides a reasonable approximation.