Calculus II: Definite Integrals & U-Substitution Review
Introduction to Definite Integrals
- 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).
Computing Integrals: The Fundamental Theorem of Calculus
- 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).
Integration Technique: U-Substitution