The online integral calculator provides a quick and accurate numerical solution for definite integrals, simplifying complex mathematical analysis for students, engineers, and researchers.
Online Definite Integral Calculator
Calculated Integral Value:
The detailed calculation steps will appear here upon successful calculation.
Online Integral Calculator Formula:
The definite integral is defined as the signed area of the region bounded by the graph of the function $f(x)$, the x-axis, and the vertical lines $x=a$ and $x=b$.
$$ \text{Integral} = \int_{a}^{b} f(x) \, dx $$
For numerical approximation, this calculator employs Simpson’s Rule:
$$ \int_{a}^{b} f(x) \, dx \approx \frac{h}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + \dots + 4f(x_{n-1}) + f(x_n)] $$
Source: Wolfram MathWorld – Definite Integral | Britannica – Simpson’s Rule
Variables:
- Function to Integrate ($f(x)$): The mathematical expression that needs to be integrated with respect to $x$. This calculator supports standard mathematical operations and functions (e.g.,
x**2,Math.sin(x)). - Lower Bound ($a$): The starting point of the integration interval on the x-axis.
- Upper Bound ($b$): The ending point of the integration interval on the x-axis.
Related Calculators:
What is Online Integral Calculator?
An online integral calculator is a digital tool designed to compute definite or indefinite integrals of functions. This particular calculator focuses on definite integrals, which represent the accumulation of quantities over a specific interval. Integration is one of the two main concepts of calculus, alongside differentiation.
The process of finding an integral, known as integration, is essentially the reverse of differentiation. While differentiation finds the rate of change of a function, integration finds the total sum or area under the curve defined by the function. Numerical integration, as used here (Simpson’s Rule), is often necessary when an analytical solution is difficult or impossible to find.
Using a calculator like this saves significant time and reduces the risk of human error in complex calculations involving polynomials, trigonometric functions, or exponential terms over defined limits.
How to Calculate Definite Integral (Example):
Let’s calculate the integral of $f(x) = x^2$ from $a=0$ to $b=3$.
- Identify the Function and Bounds: Function $f(x) = x^2$, Lower Bound $a=0$, Upper Bound $b=3$.
- Select an Integration Method: For manual calculation, the Power Rule states $\int x^n dx = \frac{x^{n+1}}{n+1} + C$. For numerical methods like Simpson’s Rule, a large number of subintervals (N) is chosen.
- Apply the Fundamental Theorem (Manual Check): $$\int_{0}^{3} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{3} = \frac{3^3}{3} – \frac{0^3}{3} = 9 – 0 = 9$$
- Use the Calculator: Input
x**2for Function,0for Lower Bound, and3for Upper Bound. Click ‘Calculate’. The calculator should return a value extremely close to 9.0.
Frequently Asked Questions (FAQ):
A definite integral has specified limits ($a$ and $b$) and results in a single numerical value (the area under the curve). An indefinite integral, or antiderivative, is a family of functions that results from reversing the differentiation process, always including the constant of integration ($+C$).
What is Simpson’s Rule?Simpson’s Rule is a numerical method for approximating the definite integral. It works by approximating the integrand with a sequence of parabolic arcs, which generally yields a highly accurate result, especially compared to the simpler Trapezoidal Rule.
What does the calculator use for the variable $x$?The calculator assumes the variable of integration is $x$. When you input your function, you must use $x$ as the variable (e.g., x**2, 2*x, Math.exp(x)).
Yes. The function input is executed using JavaScript’s Math object, allowing you to use functions like Math.sin(x), Math.cos(x), Math.exp(x), and Math.log(x).