Calculus Volume 2 Study Notes: Chapter 1 Integration

Introduction to Integration

Integration is a fundamental pillar of calculus used to determine quantities such as area, distance, and volume from rates of change.
The concept of integration often starts with the problem of finding the area of an irregular region. For example, calculate the distance an iceboat travels if its velocity is a known variable function over time.
The theoretical foundation involves approximating areas using simpler geometric shapes (like rectangles) and taking the mathematical limit as the number of shapes increases to infinity.

Sigma (Summation) Notation

Sigma notation provides a compact way to represent the sum of a long string of terms using the Greek capital letter $\Sigma$ .
The standard form is $\sum_{i=1}^{n} a_i$ , where:
- $i$ is the index of summation (or dummy variable).
- $1$ is the lower limit of the sum.
- $n$ is the upper limit of the sum.
- $a_i$ describes the general term for each index value.
Properties of Sigma Notation (where $c$ is a constant and $n, m$ are integers):
1. Constant Sum: $\sum_{i=1}^{n} c = nc$
2. Constant Multiplier: $\sum_{i=1}^{n} c a_i = c \sum_{i=1}^{n} a_i$
3. Sum Rule: $\sum_{i=1}^{n} (a_i + b_i) = \sum_{i=1}^{n} a_i + \sum_{i=1}^{n} b_i$
4. Difference Rule: $\sum_{i=1}^{n} (a_i - b_i) = \sum_{i=1}^{n} a_i - \sum_{i=1}^{n} b_i$
5. Splitting a Sum: $\sum_{i=1}^{n} a_i = \sum_{i=1}^{m} a_i + \sum_{i=m+1}^{n} a_i$ , for 1 \le m < n .
Power Sum Formulas:
- Sum of the first $n$ integers: $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$
- Sum of consecutive squares: $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$
- Sum of consecutive cubes: $\sum_{i=1}^{n} i^3 = \frac{n^2(n+1)^2}{4}$

Approximating Area Under a Curve

To calculate the area $A$ bounded by a continuous, nonnegative function $f(x)$ and the $x$ -axis on the interval $[a, b]$ , use a partition $P$ .
A partition is a set of points $P = \{x_0, x_1, \dots, x_n\}$ such that a = x_0 < x_1 < \dots < x_n = b .
In a regular partition, each subinterval has an equal width: $\Delta x = \frac{b-a}{n}$ . The points are defined by $x_i = x_0 + i \Delta x$ .
Approximation Methods:
- Left-Endpoint Approximation ( $L_n$ ): Uses the function value at the left of each subinterval for height. $L_n = \sum_{i=1}^{n} f(x_{i-1}) \Delta x$
- Right-Endpoint Approximation ( $R_n$ ): Uses the function value at the right of each subinterval for height. $R_n = \sum_{i=1}^{n} f(x_i) \Delta x$
- As $n$ (the number of rectangles) increases, both $L_n$ and $R_n$ produce more accurate estimates of the actual area.

Riemann Sums and the Definition of Area

A Riemann Sum generalizes the rectangle approximation by allowing the height to be determined by any point $x_i^*$ within each subinterval $[x_{i-1}, x_i]$ .
Definition: For a function $f(x)$ on $[a, b]$ with a regular partition, the Riemann sum is $\sum_{i=1}^{n} f(x_i^*) \Delta x$ .
Upper and Lower Sums:
- An Upper Sum is achieved when $x_i^*$ is chosen to be the location of the maximum function value in each subinterval.
- A Lower Sum is achieved when $x_i^*$ is chosen to be the location of the minimum function value in each subinterval.
Area Definition: The area $A$ under the curve $y = f(x)$ is the limit of the Riemann sums as the number of subintervals goes to infinity: $A = \lim_{n \rightarrow \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x$ .

The Definite Integral

The Definite Integral generalizes area to include functions that may be negative or discontinuous.
Formal Definition: $\int_{a}^{b} f(x) \, dx = \lim_{n \rightarrow \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x$ . If the limit exists, $f(x)$ is integrable.
Terminology:
- $\int$ is the integral symbol (an elongated S for summation).
- $a$ and $b$ are the limits of integration (lower and upper).
- $f(x)$ is the integrand.
- $x$ is the variable of integration (a dummy variable).
Integrability: Every continuous function on a closed interval $[a, b]$ is integrable. Functions with a finite number of jump discontinuities are also integrable.

Net Signed Area and Total Area

Net Signed Area: The definite integral calculates the area above the $x$ -axis minus the area below the $x$ -axis.
- $\int_{a}^{b} f(x) \, dx = A_1 - A_2$ , where $A_1$ is area above and $A_2$ is area below.
Total Area: To find the total area regardless of the $x$ -axis position, integrate the absolute value.
- \text{Total Area} = \int_{a}^{b} |f(x)| \, dx = A_1 + A_2
Application to Motion:
- Displacement is the integral of velocity: $\int_{a}^{b} v(t) \, dt$ .
- Total distance traveled is the integral of the speed: $\int_{a}^{b} |v(t)| \, dt$ .

Properties of the Definite Integral

Zero Length: $\int_{a}^{a} f(x) \, dx = 0$
Reversed Limits: $\int_{b}^{a} f(x) \, dx = - \int_{a}^{b} f(x) \, dx$
Sum/Difference: $\int_{a}^{b} [f(x) \pm g(x)] \, dx = \int_{a}^{b} f(x) \, dx \pm \int_{a}^{b} g(x) \, dx$
Constant Multiplier: $\int_{a}^{b} c f(x) \, dx = c \int_{a}^{b} f(x) \, dx$
Interval Addition: $\int_{a}^{b} f(x) \, dx = \int_{a}^{c} f(x) \, dx + \int_{c}^{b} f(x) \, dx$ (Valid for any $c$ where the function is integrable).

Comparison Theorem

If $f(x) \ge 0$ on $[a, b]$ , then $\int_{a}^{b} f(x) \, dx \ge 0$ .
If $f(x) \ge g(x)$ on $[a, b]$ , then $\int_{a}^{b} f(x) \, dx \ge \int_{a}^{b} g(x) \, dx$ .
Bound Property: If $m \le f(x) \le M$ on $[a, b]$ , then $m(b-a) \le \int_{a}^{b} f(x) \, dx \le M(b-a)$ .

Average Value of a Function

The Average Value of a continuous function $f(x)$ on $[a, b]$ is defined as:
- $f_{ave} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx$

Mean Value Theorem for Integrals

Theorem: If $f(x)$ is continuous on $[a, b]$ , there exists at least one point $c$ in $[a, b]$ such that $f(c) = f_{ave}$ .
This means the function achieves its average value at least once in the interval: $\int_{a}^{b} f(x) \, dx = f(c)(b-a)$ .

Fundamental Theorem of Calculus (FTC)

Part 1: Establishes that differentiation and integration are inverse processes.
- If $F(x) = \int_{a}^{x} f(t) \, dt$ , then $F'(x) = f(x)$ .
- Chain Rule Extension: If the upper limit is a function $g(x)$ , then $\frac{d}{dx} \int_{a}^{g(x)} f(t) \, dt = f(g(x)) g'(x)$ .
Part 2 (The Evaluation Theorem): Provides a method to calculate definite integrals without limits.
- $\int_{a}^{b} f(x) \, dx = F(b) - F(a)$ , where $F(x)$ is any antiderivative of $f(x)$ .
- Notation: $F(x) \big|_a^b = F(b) - F(a)$ .

The Net Change Theorem

Theorem: The integral of a rate of change is the net change in the quantity.
- $F(b) = F(a) + \int_{a}^{b} F'(x) \, dx$
Common applications include finding the volume of gas consumed by a boat, the population growth of breeding pairs (e.g., bald eagles), or the energy used in a city.

Symmetry in Integration: Even and Odd Functions

Even Functions: If $f(-x) = f(x)$ , then $\int_{-a}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx$ .
Odd Functions: If $f(-x) = -f(x)$ , then $\int_{-a}^{a} f(x) \, dx = 0$ .

Integration by Substitution

Substitution Rule (u-substitution): Used to reverse the chain rule.
Steps for Indefinite Integrals:
1. Choose $u = g(x)$ such that its derivative $g'(x)$ is present in the integrand.
2. Calculate $du = g'(x) \, dx$ .
3. Substitute $u$ and $du$ into the integral.
4. Evaluate the integral in terms of $u$ .
5. Substitute back the original expression $g(x)$ for $u$ .
Substitution for Definite Integrals: Requires changing the limits of integration.
- $\int_{a}^{b} f(g(x))g'(x) \, dx = \int_{g(a)}^{g(b)} f(u) \, du$
- This eliminates the need to substitute back the original variable at the end.

Questions & Discussion

Runner Velocity: If two athletes start and finish a race at the same time, they must have traveled the same distance. By the MVT, there must be at least one point where their instantaneous velocities were equal.
Speeding Tickets: Toll road authorities can use the Net Change Theorem to issue tickets by calculating average speed (distance / time) between two timestamps and entrance points.
Kepler's Laws: The second law (equal areas in equal times) implies that planets move fastest at the perihelion (closest point to the Sun) and slowest at the aphelion (farthest point).