Definite Integral Calculator with Steps

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Reviewed and Verified by David Chen, PhD (Applied Mathematics)

This calculator utilizes the robust numerical Trapezoidal Rule for high accuracy in definite integration tasks.

Welcome to the definitive tool for solving integrals! Use this calculator to instantly find the value of a definite integral over a specified interval, complete with clear, step-by-step numerical calculation details.

Definite Integral Calculator with Steps

Calculated Definite Integral Value:

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Definite Integral Calculator with Steps Formula

This calculator uses the Trapezoidal Rule for numerical approximation of the definite integral:

$$ \int_a^b f(x) \, dx \approx \frac{h}{2} [f(x_0) + 2\sum_{i=1}^{N-1} f(x_i) + f(x_N)] $$ where $h = \frac{b-a}{N}$ is the width of each subinterval.

Formula Source: Wikipedia – Trapezoidal Rule, Wolfram MathWorld

Variables Explained

The following variables are required to compute the definite integral:

  • Function $f(x)$ (Text): The mathematical expression to be integrated. Must use ‘x’ as the variable and standard JavaScript math syntax (e.g., Math.pow(x, 2), Math.sin(x)). The calculator automatically handles the common x^n syntax.
  • Lower Limit ($a$): The starting point of the integration interval.
  • Upper Limit ($b$): The ending point of the integration interval.
  • Number of Subintervals ($N$): The count of segments used for the numerical approximation. A higher $N$ yields a more accurate result. Must be a positive integer ($N>1$).

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What is Definite Integral Calculation?

The definite integral, denoted as $\int_a^b f(x) \, dx$, represents the net signed area between the graph of the function $f(x)$ and the x-axis, from $x=a$ (the lower limit) to $x=b$ (the upper limit). It is a fundamental concept in calculus used to find areas, volumes, averages, and accumulated change over an interval.

While the Fundamental Theorem of Calculus allows for exact (analytical) solutions for many functions, many complex functions do not have simple antiderivatives. In these cases, numerical integration methods, such as the Trapezoidal Rule used in this calculator, provide highly accurate approximations by dividing the area under the curve into many small, manageable shapes (trapezoids) and summing their areas.

The value returned by the definite integral is a single number, representing the quantity accumulated by the function over the given interval.

How to Calculate a Definite Integral (Example)

Let’s use the calculator’s method (Trapezoidal Rule) to find the integral of $f(x) = x^2$ from $a=0$ to $b=4$ using $N=4$ subintervals.

  1. Determine the Step Size ($h$): $h = \frac{b-a}{N} = \frac{4-0}{4} = 1$.
  2. Identify the $x_i$ points: The points are $x_0=0$, $x_1=1$, $x_2=2$, $x_3=3$, and $x_4=4$.
  3. Evaluate $f(x_i)$ at each point:
    • $f(x_0) = f(0) = 0^2 = 0$
    • $f(x_1) = f(1) = 1^2 = 1$
    • $f(x_2) = f(2) = 2^2 = 4$
    • $f(x_3) = f(3) = 3^2 = 9$
    • $f(x_4) = f(4) = 4^2 = 16$
  4. Apply the Trapezoidal Formula: $$ \text{Integral} \approx \frac{h}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + f(x_4)] $$ $$ \text{Integral} \approx \frac{1}{2} [0 + 2(1) + 2(4) + 2(9) + 16] $$ $$ \text{Integral} \approx \frac{1}{2} [0 + 2 + 8 + 18 + 16] = \frac{1}{2} [44] = 22 $$

(The exact answer is $21.333…$, showing that even with a small $N$, the numerical method is a close approximation.)

Frequently Asked Questions (FAQ)

Is the Trapezoidal Rule perfectly accurate?
No, the Trapezoidal Rule is a numerical approximation, not an exact solution. Its accuracy improves significantly as the number of subintervals ($N$) increases, which is why we recommend using a large $N$ (e.g., 100 or 1000) for real-world calculations.

What is the maximum $N$ (subintervals) I should use?
While a higher $N$ gives better accuracy, using excessively large numbers (e.g., over 1,000,000) can slow down the calculation and may introduce minor floating-point errors. For most purposes, $N=1000$ provides an excellent balance of speed and precision.

How should I format the function $f(x)$ input?
Use ‘x’ as the variable. Use standard math syntax: multiplication is `*` (e.g., `2*x`), standard math functions are prefixed with `Math.` (e.g., `Math.sin(x)`, `Math.log(x)`), and exponents can be written as `x^2` or `Math.pow(x, 2)`.

Why is the calculation not working when I press the button?
Check the error message above the button. Common issues include invalid function syntax (e.g., forgetting a `*` for multiplication), non-numeric limits, or using an $N$ value less than 2. Also, ensure the function is defined across the entire interval $[a, b]$.

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