Overview
This lecture explains the difference between indefinite and definite integrals, introduces the Fundamental Theorem of Calculus Part 2, and demonstrates evaluating definite integrals through several examples.
Indefinite vs. Definite Integrals
- Indefinite integrals do not have upper or lower limits and yield a general antiderivative plus a constant, C.
- Definite integrals have specified upper (b) and lower (a) limits and evaluate to a specific number.
Key Parts of Integral Notation
- The integrand is the function being integrated (f(x)).
- dx indicates the variable of integration.
- The integral sign represents a limit of sums.
Fundamental Theorem of Calculus Part 2 (FTC2)
- FTC2 states: ∫ from a to b of f(x) dx = F(b) - F(a), where F is the antiderivative of f.
Example 1: Constant Integrand
- ∫ from 2 to 5 of 8 dx: antiderivative is 8x, so evaluate 8×5 - 8×2 = 40 - 16 = 24.
Example 2: Linear Integrand
- ∫ from 1 to 4 of (5x - 4) dx:
- Antiderivative is (5/2)x² - 4x.
- Plug in 4 and 1, compute difference: result is 51/2, or 25.5.
Example 3: Negative Powers
- ∫ from -3 to 4 of 8/x³ dx:
- Rewrite as 8x^-3; antiderivative is -4x^-2 = -4/x².
- Compute -1/4 + 4/9 = 7/36.
Example 4: Natural Logarithm
- ∫ from 1 to e of (5/x) dx:
- Antiderivative is 5ln(x).
- Evaluate 5ln(e) - 5ln(1) = 5×1 - 5×0 = 5.
Example 5: Roots and Fractional Exponents
- ∫ from 4 to 9 of 1/√x dx:
- Rewrite as x^(-1/2); antiderivative is 2x^(1/2).
- Evaluate 2√9 - 2√4 = 6 - 4 = 2.
Key Terms & Definitions
- Definite Integral — Integral with upper and lower limits, yields a specific value.
- Indefinite Integral — Integral without limits, yields a general antiderivative plus C.
- Integrand — The function being integrated.
- Antiderivative — A function whose derivative is the original function.
- Fundamental Theorem of Calculus (Part 2) — Links definite integrals to evaluating antiderivatives at endpoints.
Action Items / Next Steps
- Practice evaluating definite integrals using the power rule and FTC2.
- Review antiderivative rules for polynomials and logarithmic functions.