5.2 The Definite Integral

Definite integrals are powerful tools for calculating areas and accumulating quantities. They combine an integrand, 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 limit 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 $x$
• Definite integral notation $\int_a^b f(x) dx$ combines these components
• Riemann sum approximation $\sum_{i=1}^n f(x_i^*) \Delta x$ partitions the interval into $n$ subintervals of width $\Delta x$ and evaluates the function at sample points $x_i^*$ 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
• Continuity of a function on the interval [a, b] guarantees its integrability

Definite integrals as net area

• Definite integral $\int_a^b f(x) dx$ represents the net area between the graph of $f(x)$ and the $x$-axis over the interval $[a, b]$
• For non-negative functions ($f(x) \geq 0$), the definite integral equals the total area 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

1. Fundamental Theorem of Calculus, Part 1: If $F(x)$ is an antiderivative of $f(x)$, then $\int_a^b f(x) dx = F(b) - F(a)$, relating the definite integral to the antiderivative (indefinite integral)
2. Properties of definite integrals:
   • Linearity: $\int_a^b [cf(x) + dg(x)] dx = c\int_a^b f(x) dx + d\int_a^b g(x) dx$ for constants $c$ and $d$
   • Additivity: $\int_a^b f(x) dx + \int_b^c f(x) dx = \int_a^c f(x) dx$, allowing the interval to be split into smaller subintervals
   • Symmetry: $\int_a^b f(x) dx = -\int_b^a f(x) dx$, reversing the limits of integration changes the sign of the definite integral
3. 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 $\frac{1}{b-a} \int_a^b f(x) dx$ 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)