Definite integrals are powerful tools for calculating areas and accumulating quantities. They combine an , limits, and a differential to represent the area under a curve or between curves.
Evaluating definite integrals involves using the Fundamental Theorem of Calculus and various properties. The average value of a function can be found using definite integrals, providing insights into a function's behavior over an interval.
The Definite Integral
Components of definite integrals
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Integrand f(x) represents the function being integrated over a specific interval
Lower of integration a defines the starting point of the interval
Upper limit of integration b defines the endpoint of the interval
Differential dx indicates integration with respect to the variable x
notation ∫abf(x)dx combines these components
Riemann sum approximation ∑i=1nf(xi∗)Δx partitions the interval into n subintervals of width Δx and evaluates the function at sample points xi∗ within each subinterval
Definite integral represents the limit of the Riemann sum as the number of subintervals approaches infinity, providing a precise value for the area under the curve
Integrability of functions
Function f(x) is integrable on the interval [a,b] if the limit of the Riemann sum exists and converges to a unique value as the number of subintervals approaches infinity
Integrability requires left and right Riemann sums to approach the same value, regardless of the choice of sample points
Ensures the definite integral is well-defined and can be evaluated consistently
Non-integrable functions may have definite integrals that do not exist or yield different values depending on the sampling method, leading to ambiguity and inconsistency
of a function on the interval [a, b] guarantees its integrability
Definite integrals as net area
Definite integral ∫abf(x)dx represents the net area between the graph of f(x) and the x-axis over the interval [a,b]
For non-negative functions (f(x)≥0), the definite integral equals the under the curve
For functions that change sign, the definite integral calculates the area above the x-axis minus the area below the x-axis, resulting in the net area
Geometric interpretation provides visual insight into the meaning and properties of definite integrals
Allows for the calculation of areas bounded by curves, even for irregular shapes or functions lacking explicit formulas
Fundamental concepts of definite integrals
Accumulation: The definite integral represents the accumulation of a quantity over an interval
Limit: The definite integral is defined as the limit of Riemann sums as the number of subintervals approaches infinity
Georg Friedrich Bernhard Riemann: Developed the concept of Riemann sums, which form the basis for defining and evaluating definite integrals
Evaluating and Interpreting Definite Integrals
Evaluation methods for definite integrals
Fundamental Theorem of Calculus, Part 1: If F(x) is an antiderivative of f(x), then ∫abf(x)dx=F(b)−F(a), relating the definite integral to the antiderivative (indefinite integral)
Properties of definite integrals:
Linearity: ∫ab[cf(x)+dg(x)]dx=c∫abf(x)dx+d∫abg(x)dx for constants c and d
Additivity: ∫abf(x)dx+∫bcf(x)dx=∫acf(x)dx, allowing the interval to be split into smaller subintervals
: ∫abf(x)dx=−∫baf(x)dx, reversing the changes the sign of the definite integral
Integration rules (power rule, trigonometric substitution, integration by parts) simplify the evaluation of definite integrals for common function types
Average value through definite integrals
Average value formula b−a1∫abf(x)dx calculates the average height of the function f(x) over the interval [a,b]
Divides the definite integral (total area) by the width of the interval (b−a) to obtain the average value
Physical interpretations:
Average velocity, acceleration, or force over a time interval (physics)
Average cost, revenue, or profit over a production interval (economics)
Provides a concise summary of the function's behavior and central tendency within the given interval
Useful for analyzing and comparing functions in various contexts (engineering, social sciences)