Definite Integral Calculator
Calculate definite integral instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Formula
Integral from a to b of f(x) dx = F(b) - F(a)
Where F(x) is the antiderivative of f(x). For numerical approximation, Simpson rule uses the formula (h/3)[f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + ... + f(xn)], where h = (b-a)/n is the step size.
Worked Examples
Example 1: Polynomial Integration
Problem: Compute the definite integral of f(x) = x^2 + 2x + 1 from x = 0 to x = 4.
Solution: Antiderivative: F(x) = x^3/3 + x^2 + x\nF(4) = 64/3 + 16 + 4 = 21.333 + 16 + 4 = 41.333\nF(0) = 0\nExact integral = 41.333...\nSimpson rule (1000 segments): 41.33333333\nTrapezoidal rule (1000 segments): 41.33333333
Result: Exact: 41.33333333 | Simpson: 41.33333333 | Average Value: 10.33333
Example 2: Exponential Integral
Problem: Compute the integral of f(x) = 2*e^(0.5x) from x = 0 to x = 4.
Solution: Antiderivative: F(x) = (2/0.5)*e^(0.5x) = 4*e^(0.5x)\nF(4) = 4*e^2 = 4 * 7.389 = 29.556\nF(0) = 4*e^0 = 4\nExact integral = 29.556 - 4 = 25.556\nNumerical methods converge to this value with high precision.
Result: Exact: 25.55644768 | Simpson: 25.55644768 | Average Value: 6.3891
Frequently Asked Questions
What is a definite integral and what does it calculate?
A definite integral calculates the signed area between a function and the x-axis over a specific interval [a, b]. It is written as the integral from a to b of f(x) dx and represents the net accumulation of the quantity described by f(x). Unlike an indefinite integral (which produces a family of antiderivatives), a definite integral produces a single numerical value. This value can represent physical quantities like distance traveled, work done, total charge, or accumulated probability. The Fundamental Theorem of Calculus connects definite integrals to antiderivatives, stating that the integral from a to b of f(x) dx equals F(b) - F(a), where F is any antiderivative of f.
What are Riemann sums and how do they relate to the definite integral?
Riemann sums are the foundational concept behind definite integration. They approximate the area under a curve by dividing the interval into subintervals and summing the areas of rectangles whose heights are determined by function values. Left Riemann sums use the left endpoint of each subinterval, right sums use the right endpoint, and midpoint sums use the center. As the number of rectangles approaches infinity and their width approaches zero, the Riemann sum converges to the definite integral (for integrable functions). This limiting process is the formal definition of the Riemann integral. While Riemann sums are conceptually important, they converge slowly compared to higher-order methods like Simpson rule, making them impractical for precise numerical computation.
How do improper integrals differ from definite integrals?
Improper integrals extend definite integrals to handle infinite limits of integration or integrands with singularities within the interval. Type 1 improper integrals have at least one infinite limit, such as the integral from 1 to infinity of 1/x^2 dx. Type 2 improper integrals have integrands that are unbounded at some point in the interval, such as the integral from 0 to 1 of 1/sqrt(x) dx. Both types are evaluated as limits of proper definite integrals. An improper integral converges if this limit exists and is finite; otherwise it diverges. Convergence tests like the comparison test and p-test help determine whether an improper integral converges without evaluating it. Definite Integral Calculator handles proper definite integrals; for improper integrals, use a large finite bound as an approximation.
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How do I interpret the result?
Results are displayed with a label and unit to help you understand the output. Many calculators include a short explanation or classification below the result (for example, a BMI category or risk level). Refer to the worked examples section on this page for real-world context.