Definite Integral Calculator
Numerically evaluates the definite integral of a power function ax^n over a specified interval. Choose from Simpson's Rule, Trapezoidal Rule, or the Midpoint Rectangle Method for your approximation.
About this calculator
A definite integral measures the signed area under a curve between two limits. This calculator approximates ∫[a to b] c·xⁿ dx using three classic numerical methods. Simpson's Rule uses the formula S = (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + … + f(xₙ)] and is generally the most accurate. The Trapezoidal Rule approximates the area as a series of trapezoids: T = h·[f(x₀)/2 + f(x₁) + … + f(xₙ)/2]. The Midpoint Rectangle Method evaluates the function at the center of each sub-interval: M = h·Σf(xᵢ + h/2). In all three cases, h = (upperLimit − lowerLimit) / intervals. Increasing the number of intervals improves accuracy for any method.
How to use
Suppose you want to integrate f(x) = 2x² from 0 to 3 using Simpson's Rule with 4 intervals. Set coefficient = 2, exponent = 2, lower limit = 0, upper limit = 3, intervals = 4. Then h = (3 − 0)/4 = 0.75. Evaluate f at x = 0, 0.75, 1.5, 2.25, 3: values are 0, 1.125, 4.5, 10.125, 18. Apply S = (0.75/3)[0 + 4(1.125) + 2(4.5) + 4(10.125) + 18] = 0.25 × [0 + 4.5 + 9 + 40.5 + 18] = 0.25 × 72 = 18. The exact answer is also 18, confirming Simpson's Rule is exact for polynomials up to degree 3.
Frequently asked questions
What is the difference between Simpson's Rule and the Trapezoidal Rule for definite integrals?
The Trapezoidal Rule approximates the area under a curve by connecting function values with straight lines, forming trapezoids. Simpson's Rule instead fits parabolic arcs through sets of three points, which captures curvature much better. As a result, Simpson's Rule is generally more accurate for the same number of intervals, especially for smooth functions. For polynomials of degree 3 or lower, Simpson's Rule is exact. The Trapezoidal Rule is simpler to implement but requires far more intervals to achieve comparable precision.
How many intervals should I use for an accurate definite integral approximation?
The required number of intervals depends on how rapidly the function changes and which method you choose. For Simpson's Rule, even 4–10 intervals often give excellent accuracy for smooth power functions. The Trapezoidal Rule converges more slowly, so you may need 50–100 intervals for similar precision. A practical strategy is to double the interval count and check if the result changes significantly — if not, you have converged. For highly oscillatory or steeply changing functions, more intervals are always safer.
Why does the definite integral give the area under the curve?
The definite integral ∫[a to b] f(x) dx is defined as the limit of a sum of infinitely thin rectangles, each of width dx and height f(x). As the width shrinks to zero, this Riemann sum converges to the exact signed area between the curve and the x-axis. Regions where f(x) > 0 contribute positive area, while regions where f(x) < 0 contribute negative area. Numerical methods like Simpson's and the Trapezoidal Rule approximate this limit using a finite number of sub-intervals, trading a small error for computational feasibility.