Integral Calculator
Calculate definite integrals numerically with our free Integral Calculator. Enter a function of x, specify the lower and upper bounds, and choose a numerical method — Simpson's rule, Trapezoidal rule, or Riemann sum. See the result, the area under the curve, and a visual representation.
What Is Numerical Integration?
Numerical integration is the process of approximating the definite integral of a function using computational methods. It is used when the antiderivative is difficult or impossible to find analytically. The definite integral ∫ab f(x) dx represents the area under the curve of f(x) from x = a to x = b.
This calculator provides three common numerical integration methods:
- Simpson's Rule: Uses quadratic polynomials to approximate the function. The most accurate of the three methods for smooth functions.
- Trapezoidal Rule: Uses trapezoids to approximate the area under the curve. Simple and effective for many functions.
- Riemann Sum: Uses rectangles to approximate the area. You can choose left, right, or midpoint rectangles.
The accuracy of each method increases with the number of subintervals (n). This calculator allows you to adjust n to balance speed and precision.
How the Integral Calculator Works
The calculator uses the math.js library to parse and evaluate mathematical expressions. Here's how each method works:
Simpson's Rule (n even):
∫ f(x) dx ≈ (h/3) × [f(x₀) + f(xn) + 4∑f(xodd) + 2∑f(xeven)]
Trapezoidal Rule:
∫ f(x) dx ≈ (h/2) × [f(x₀) + f(xn) + 2∑f(xi)]
Riemann Sum (Midpoint):
∫ f(x) dx ≈ h × ∑ f((xi-1 + xi)/2)
The calculator also visualizes the function and the area under the curve, making it easy to understand the geometric interpretation of the integral.
When to Use Different Methods
- Simpson's Rule: Best for smooth, continuous functions. Requires n to be even. Provides the most accurate results with fewer subintervals.
- Trapezoidal Rule: Good for most functions. Simpler than Simpson's but less accurate for the same number of subintervals.
- Riemann Sum: The simplest method. Best for understanding the fundamental concept of integration. Use midpoint for better accuracy than left or right endpoints.
❓ Integral Calculator FAQ
What is a definite integral?
A definite integral ∫ab f(x) dx represents the signed area under the curve of f(x) from x = a to x = b. It can be interpreted as the total accumulation of the function over that interval.
What is numerical integration?
Numerical integration is the process of approximating the value of a definite integral using computational methods. It is used when the antiderivative of a function is difficult or impossible to find analytically.
What is Simpson's Rule?
Simpson's Rule is a numerical integration method that uses quadratic polynomials (parabolas) to approximate the function. It is highly accurate for smooth functions and requires an even number of subintervals.
What is the Trapezoidal Rule?
The Trapezoidal Rule approximates the area under a curve by dividing it into trapezoids. It is simpler than Simpson's Rule but less accurate for the same number of subintervals.
What is a Riemann sum?
A Riemann sum approximates the area under a curve by dividing it into rectangles. The height of each rectangle is determined by the function value at a chosen point within each subinterval (left endpoint, right endpoint, or midpoint).
How does the number of subintervals affect accuracy?
Increasing the number of subintervals (n) generally increases the accuracy of the approximation. More subintervals mean smaller intervals, which allow the numerical method to better capture the shape of the function.
What is the difference between left, right, and midpoint Riemann sums?
Left Riemann sum uses the function value at the left endpoint of each subinterval. Right uses the right endpoint. Midpoint uses the value at the middle of each subinterval, which typically provides a better approximation than left or right.
What functions can I integrate with this calculator?
You can integrate any function that can be parsed by math.js. This includes polynomials (x^2, x^3), trigonometric functions (sin(x), cos(x), tan(x)), exponential and logarithmic functions (exp(x), log(x)), and combinations of these.
What is the error estimate shown in the results?
The approximate error is calculated by comparing the result with a more accurate approximation using double the number of subintervals. This gives you an idea of the accuracy of the result.
What is the fundamental theorem of calculus?
The fundamental theorem of calculus states that differentiation and integration are inverse operations. It connects the definite integral of a function to its antiderivative, allowing exact calculation of integrals when the antiderivative is known.
What is an improper integral?
An improper integral is a definite integral where either the interval of integration is infinite or the function has an infinite discontinuity within the interval. This calculator does not handle improper integrals directly.
What is the area under the curve?
The area under the curve is the geometric interpretation of the definite integral. It represents the total area between the function and the x-axis over a given interval. Areas above the x-axis are positive, and areas below are negative.
How do I enter functions in the calculator?
Use standard mathematical notation: sin(x), cos(x), tan(x), exp(x) for e^x, log(x) for natural log, sqrt(x) for square root, and ^ for exponentiation. For example: x^2, 3*sin(x), exp(-x^2).
What is the difference between indefinite and definite integrals?
An indefinite integral (antiderivative) is a function whose derivative is the original function. A definite integral calculates the area under the curve over a specific interval and produces a numerical value. This calculator computes definite integrals.
What is the error in Simpson's Rule?
The error in Simpson's Rule is proportional to (b-a)⁵ / n⁴, where n is the number of subintervals. This means Simpson's Rule converges very quickly for smooth functions — doubling n reduces the error by a factor of about 16.
What is the error in the Trapezoidal Rule?
The error in the Trapezoidal Rule is proportional to (b-a)³ / n². Doubling n reduces the error by a factor of about 4, which is slower convergence than Simpson's Rule.
Why is Simpson's Rule more accurate than the Trapezoidal Rule?
Simpson's Rule uses quadratic polynomials to approximate the function, which better captures curvature than the straight lines used by the Trapezoidal Rule. This results in significantly higher accuracy for the same number of subintervals.
Can this calculator integrate functions with singularities?
This calculator uses numerical methods that assume the function is well-behaved on the interval. If the function has a singularity (infinite value) within the interval, the results may be inaccurate. Consider splitting the integral around the singularity.