Calculus II: Definite Integrals & U-Substitution Review

Definition: The definite integral computes directed or signed area between a curve and the x-axis.
- Area above the x-axis is counted as positive.
- Area below the x-axis is counted as negative.
- It represents the net total amount of area over a given interval.
Key Concept: Net Change: More importantly, definite integrals compute net changes for a quantity.
- Positive area indicates an increase in value.
- Negative area indicates a decrease in value.
Example (Bank Account):
- Let $F(t)$ be the amount of money in a bank account at time $t$ (e.g., year).
- Its derivative, $F'(t)$ (let's call it $f(t)$ ), represents the rate of change of money in the account per year.
- The definite integral $\int_0^y f(t) dt$ calculates the net change in the bank account money from year $0$ to year $y$ (the amount it went up or down by).

Geometric Approach: Integrals can be approximated or computed by breaking the area under the curve into geometric shapes (rectangles, triangles, semicircles) and summing their areas, accounting for signs.
Fundamental Theorem of Calculus (FTC): This theorem provides a more efficient way to compute definite integrals.
- If $f(x)$ is a continuous function and $F(x)$ is any antiderivative of $f(x)$ (meaning $F'(x) = f(x)$ ), then the definite integral from $a$ to $b$ is given by: $\int_a^b f(x) dx = F(b) - F(a)$ .
- This is known as the net change theorem. It states that the net change of $F(x)$ from $a$ to $b$ is the integral of its rate of change over that interval.
Antiderivatives and Indefinite Integrals:
- An antiderivative of $f(x)$ is a function $F(x)$ such that $F'(x) = f(x)$ .
- The indefinite integral $\int f(x) dx$ asks for the most general antiderivative of $f(x)$ , which includes an arbitrary constant $+C$ (e.g., $F(x) + C$ ).
Common Antiderivatives (Memorization): A list of standard antiderivatives is expected to be memorized (often provided on flashcards).
- Special Case to Memorize: $\int \ln x dx = x \ln x - x + C$ (its derivation will be covered later).