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)
Key Terms to Review (18)
Average value of the function: The average value of a function over an interval $[a, b]$ is given by $\frac{1}{b-a} \int_a^b f(x) \, dx$. It represents the mean value of all function outputs in that interval.
Constant multiple law for limits: The Constant Multiple Law for limits states that the limit of a constant multiplied by a function is equal to the constant multiplied by the limit of the function. Mathematically, if $\lim_{{x \to c}} f(x) = L$, then $\lim_{{x \to c}} [k \cdot f(x)] = k \cdot L$ where $k$ is a constant.
Continuity: Continuity is a fundamental concept in calculus that describes the smoothness and uninterrupted nature of a function. It is a crucial property that allows for the application of calculus techniques and the study of limits, derivatives, and integrals.
Continuity over an interval: Continuity over an interval means that a function is continuous at every point within a given interval. This implies that the function has no breaks, jumps, or holes in that interval.
Definite integral: A definite integral is the evaluation of the integral of a function over a specific interval, yielding a real number that represents the net area under the curve between two points.
Displacement: Displacement is the net change in position of an object, calculated as the difference between the final and initial positions. It can be determined using definite integrals in calculus.
Dummy variable: A dummy variable is a placeholder variable used in integration to represent the variable of integration. It has no impact on the final value of the definite integral.
Fave: The definite integral of a function over an interval $[a, b]$ represents the net area under the curve of the function from $x = a$ to $x = b$. It is denoted by $\int_{a}^{b} f(x) \, dx$ and is calculated as the limit of Riemann sums.
Integrable function: An integrable function is a function for which the definite integral over a specified interval exists and is finite. This concept ensures that the area under the curve of the function can be measured.
Integrand: The integrand is the function being integrated in an integral. It appears inside the integral symbol and is the main focus of the integration process.
Leibniz: Gottfried Wilhelm Leibniz was a German mathematician and philosopher who independently developed calculus around the same time as Isaac Newton. His notation for derivatives and integrals is widely used today.
Limit: In mathematics, the limit of a function is a fundamental concept that describes the behavior of a function as its input approaches a particular value. It is a crucial notion that underpins the foundations of calculus and serves as a building block for understanding more advanced topics in the field.
Limits of integration: Limits of integration are the values that define the interval over which a definite integral is evaluated. They are typically represented as the lower limit $a$ and the upper limit $b$ in the integral notation $\int_{a}^{b}$.
Net signed area: Net signed area is the total area between a curve and the x-axis, accounting for areas where the curve is below the x-axis as negative. It is computed using definite integrals and can result in a positive, negative, or zero value.
Symmetry: Symmetry refers to the balanced and proportional arrangement of elements in a design or object. It describes the quality of being made up of exactly similar parts facing each other or around an axis, center, or edge.
Symmetry about the origin: A function is symmetric about the origin if rotating its graph 180 degrees around the origin does not change the graph. Mathematically, this means $f(-x) = -f(x)$ for all $x$ in the domain of $f$.
Total area: Total area is the absolute value of the sum of the areas between a function and the x-axis over a given interval. It accounts for all regions, both above and below the x-axis, by considering their absolute values.
Variable of integration: A variable of integration is the variable with respect to which an integral is computed. It is typically represented as 'dx' in the integral notation.