Definite integrals are powerful tools for calculating areas and accumulating quantities. They use limits and sums to find the area between a curve and the x-axis over a specific interval, considering both positive and negative contributions.

Evaluating definite integrals involves various techniques, from geometric properties to integration rules. These methods allow us to solve real-world problems in physics, engineering, and economics, like finding volumes, arc lengths, and average values of functions.

The Definite Integral

Components of definite integrals

• Definite integral notation $\int_a^b f(x) dx$ represents the signed area between the curve $y = f(x)$ and the $x$-axis from $x = a$ to $x = b$
  • Integrand $f(x)$ specifies the function being integrated
  • Lower limit of integration $a$ marks the starting point of the interval
  • Upper limit of integration $b$ indicates the endpoint of the interval
  • Differential $dx$ signifies integration with respect to the variable $x$
• Signed area interpretation distinguishes between positive areas above the $x$-axis and negative areas below the $x$-axis
  • Areas above the $x$-axis contribute positively to the definite integral
  • Areas below the $x$-axis contribute negatively to the definite integral

Integrability of functions

• Function $f(x)$ is integrable on the interval $[a, b]$ if the definite integral $\int_a^b f(x) dx$ exists and has a finite value
• Integrability requires the function to be bounded on the interval $[a, b]$
  • Bounded functions have a minimum value $m$ and a maximum value $M$ such that $m \leq f(x) \leq M$ for all $x$ in $[a, b]$
• Integrability ensures the definite integral can be evaluated and yields a meaningful result ($area$, $volume$, $work$)
• Continuity of a function on $[a, b]$ guarantees its integrability

Definite integrals as net area

• Definite integral $\int_a^b f(x) dx$ computes the net area between the curve $y = f(x)$ and the $x$-axis from $x = a$ to $x = b$
  • Net area sums the positive areas above the $x$-axis and the negative areas below the $x$-axis
• Non-negative functions ($f(x) \geq 0$) on $[a, b]$ have a definite integral equal to the area between the curve and the $x$-axis
• Non-positive functions ($f(x) \leq 0$) on $[a, b]$ have a definite integral equal to the negative of the area between the curve and the $x$-axis

Riemann Sums and Limit Process

• Definite integrals can be understood as the limit of Riemann sums
• Process involves:
  • Partitioning the interval $[a, b]$ into subintervals
  • Approximating the area using rectangles or other shapes
  • Taking the limit as the number of subintervals approaches infinity
• Summation of these approximations converges to the definite integral
• This process relates to the concept of accumulation of infinitesimal quantities

Evaluating and Applying the Definite Integral

Techniques for evaluating definite integrals

• Geometric properties simplify definite integral calculations for specific function types
  • Even functions ($f(-x) = f(x)$) with symmetric limits ($a = -b$): $\int_{-a}^a f(x) dx = 2 \int_0^a f(x) dx$
  • Odd functions ($f(-x) = -f(x)$) with symmetric limits ($a = -b$): $\int_{-a}^a f(x) dx = 0$
• Integration rules provide methods for manipulating and evaluating definite integrals
  1. Linearity combines definite integrals of scaled functions: $\int_a^b [c_1 f(x) + c_2 g(x)] dx = c_1 \int_a^b f(x) dx + c_2 \int_a^b g(x) dx$
  2. Additivity breaks definite integrals into subintervals: $\int_a^b f(x) dx + \int_b^c f(x) dx = \int_a^c f(x) dx$
  3. Fundamental Theorem of Calculus relates antiderivatives to definite integrals: $\int_a^b f(x) dx = F(b) - F(a)$ where $F'(x) = f(x)$

Average value through definite integrals

• Average value formula $\frac{1}{b-a} \int_a^b f(x) dx$ computes the arithmetic mean of function values over the interval $[a, b]$
  • Equivalent to the height of a rectangle with base $[a, b]$ and the same area as the area under the curve $y = f(x)$ on $[a, b]$
• Average value interpretation depends on the problem context ($average$ $velocity$, $average$ $density$, $average$ $cost$)

Applications of definite integrals

• Definite integrals solve real-world problems in various fields ($physics$, $engineering$, $economics$)
• Common applications include:
  • Area between curves ($region$ $bounded$ $by$ $functions$)
  • Volume of solids of revolution ($rotating$ $a$ $region$ $about$ $an$ $axis$)
  • Arc length ($distance$ $along$ $a$ $curve$)
  • Work done by a variable force ($force$ $as$ $a$ $function$ $of$ $displacement$)
  • Moments and centers of mass ($distribution$ $of$ $mass$ $in$ $an$ $object$)
  • Probability and expected value ($continuous$ $random$ $variables$)
• Problem-solving steps:
  1. Identify relevant variables and functions
  2. Set up the appropriate integral expression
  3. Evaluate the integral using suitable techniques
  4. Interpret the result in the context of the problem