Mathematics, Statistics & Geometry

Definite Integral Calculator

Evaluate definite integrals numerically using Simpson's 1/3 Rule. Supports advanced mathematical expressions, variable bounds, and high precision.

Definite Integral
66.667
Approximation MethodSimpson's 1/3 Rule
Calculation StepsFunction: f(x) = x^2 + 2*x Bounds: a = 0, b = 5 Intervals (n) = 1000, Step size (h) = 0.005000 Using Simpson's 1/3 Rule: ∫ f(x) dx ≈ (h/3) * [f(x₀) + 4Σf(x_odd) + 2Σf(x_even) + f(xₙ)] Result ≈ 66.666667

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Finding the Area Under the Curve

The Definite Integral Calculator is a powerhouse tool for calculus students and engineers. By utilizing Simpson's 1/3 Rule, it performs high-precision numerical integration to find the exact accumulated value of any function over a specified interval.

∫abf(x)dx≈h3[f(x0)+4∑f(xodd)+2∑f(xeven)+f(xn)]\begin{aligned} \int_{a}^{b} f(x) dx \approx \frac{h}{3} \left[ f(x_0) + 4\sum f(x_{odd}) + 2\sum f(x_{even}) + f(x_n) \right] \end{aligned}

Where:
a, b=
The start and end points of the interval
f(x)=
The mathematical function being integrated
h=
The width of each sub-interval

Interpreting the Results

The result of a definite integral is a single number.

  • If the curve is mostly above the x-axis, the integral will be positive.
  • If the curve is mostly below the x-axis, the integral will be negative.
  • If it is perfectly symmetrical above and below, the integral may be zero.

Applications of Integration

  • Physics: Calculating total distance traveled from a velocity function, or total work done by a variable force.
  • Probability: Finding the probability that a random variable falls within a specific range using probability density functions.
  • Finance: Determining the present value of a continuous stream of future income.
  • Geometry: Calculating the exact volume of irregular 3D solids by integrating cross-sectional areas.

Frequently Asked Questions

A definite integral represents the exact area under a curve between two specific points (the lower bound 'a' and upper bound 'b') on the x-axis.

Simpson's Rule is a numerical method for approximating integrals. Instead of drawing straight lines between points (like the trapezoidal rule), it fits parabolas across three points, resulting in highly accurate approximations.

Many functions do not have an elementary antiderivative (e.g., e^(-x²)), meaning they cannot be solved symbolically. Numerical methods can approximate the area of ANY continuous function.

Use standard programming math notation. For example: 'x^2' for x-squared, 'sin(x)' for sine, 'sqrt(x)' for square root, and 'exp(x)' for e^x.

It determines how many slices the area is cut into. A higher number of intervals (e.g., 1000) provides a more precise answer but requires more computational power.

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