Use the Integral Calculator below to quickly estimate the definite integral of a single-variable function over a specified interval using the numerical Trapezoidal Rule.
Integral Calculator
Integral Calculator Formula (Trapezoidal Rule)
This calculator uses the numerical method known as the Trapezoidal Rule to approximate the definite integral $\int_a^b f(x) dx$. This method is generally accurate when the number of intervals ($n$) is large.
$$\int_a^b f(x) dx \approx \frac{b-a}{2n} [f(x_0) + 2f(x_1) + 2f(x_2) + \dots + 2f(x_{n-1}) + f(x_n)]$$
Formula Source: Wolfram MathWorld – Trapezoidal Rule, Wikipedia – Trapezoidal Rule
Variables
The inputs required for the numerical integral approximation are:
- Function $f(x)$: The mathematical expression you want to integrate. It must be written using JavaScript syntax (e.g., use
Math.pow(x, 2)for $x^2$). - Lower Limit ($a$): The starting point of the integration interval.
- Upper Limit ($b$): The ending point of the integration interval. Must be greater than the Lower Limit ($b > a$).
- Number of Intervals ($n$): The number of trapezoids used for the approximation. A larger integer value (e.g., $n \ge 1000$) generally yields a more accurate result.
Related Calculators
What is integral calculator?
An integral calculator is a tool that computes the integral (either indefinite or definite) of a function. The integral is a fundamental concept in calculus, representing the accumulation of quantities and the area under the graph of a function. In general, integration is the inverse process of differentiation.
There are two main types of integrals: the indefinite integral (antiderivative) and the definite integral. While symbolic integral calculators provide the exact analytical solution (the formula for the antiderivative), this tool focuses on **numerical definite integration**, which provides a precise numerical value for the area under the curve between two specific points, $a$ and $b$. Numerical methods are crucial when analytical solutions are difficult or impossible to find.
How to Calculate Integral (Example)
Let’s use the Trapezoidal Rule to approximate the integral of $f(x) = x^2$ from $a=0$ to $b=1$ with $n=4$ intervals.
- Define the function and limits: $f(x) = x^2$, $a=0$, $b=1$.
- Calculate the interval width ($\Delta x$): $\Delta x = (b-a) / n = (1 – 0) / 4 = 0.25$.
- Find the partition points ($x_i$): $x_0=0, x_1=0.25, x_2=0.5, x_3=0.75, x_4=1$.
- Apply the Trapezoidal Rule: $$\text{Area} \approx \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + \dots + f(x_4)]$$ $$\text{Area} \approx \frac{0.25}{2} [0^2 + 2(0.25^2) + 2(0.5^2) + 2(0.75^2) + 1^2]$$ $$\text{Area} \approx 0.125 [0 + 0.125 + 0.5 + 1.125 + 1] = 0.125 \times 2.75 = 0.34375$$
- Result: The approximate integral is $0.34375$. (Note: The exact answer is $1/3 \approx 0.33333$). Our calculator automates this complex process for any $n$.
Frequently Asked Questions (FAQ)
What is the difference between definite and indefinite integrals?
A definite integral $\int_a^b f(x) dx$ results in a single numerical value (representing area or accumulation) between two limits, $a$ and $b$. An indefinite integral $\int f(x) dx$ results in a family of functions (the antiderivative, plus a constant $C$).
Why does this calculator use the Trapezoidal Rule?
The Trapezoidal Rule is a straightforward and effective numerical method for approximating definite integrals. It involves dividing the area under the curve into a series of trapezoids. It provides a good balance between simplicity and accuracy, especially with a large number of intervals.
What does the variable ‘n’ represent?
The variable ‘n’ is the number of subintervals (or trapezoids) used to partition the integration range $[a, b]$. A higher value of $n$ increases the resolution of the approximation, leading to greater accuracy, but also increases the calculation time.
Can I integrate functions with more than one variable?
No, this calculator is designed only for single-variable definite integrals, $\int_a^b f(x) dx$. Multivariable integrals (double or triple integrals) require more complex numerical methods and are outside the scope of this simple tool.