Overview
This lecture explains how to evaluate definite integrals by finding antiderivatives and applying limits of integration, using step-by-step examples.
Definite vs. Indefinite Integrals
- A definite integral has lower (a) and upper (b) limits of integration.
- An indefinite integral does not have specific integration limits.
- The result of a definite integral is a number; indefinite integrals result in a general function plus a constant (C).
Finding Antiderivatives
- The antiderivative (integral) of f(x) is denoted as F(x).
- To find the antiderivative of xⁿ, use xⁿ⁺¹/(n+1).
- For a constant times xⁿ, factor out the constant and then integrate xⁿ.
- For definite integrals, you do not add the constant of integration (C).
Evaluating a Definite Integral: Step-by-Step
- Integrate each term in the function separately.
- For example, ∫8x³dx = 8x⁴/4; ∫3x²dx = 3x³/3; ∫6xdx = 6x²/2.
- Simplify each term: 8/4 = 2; 3/3 = 1; 6/2 = 3.
- Combine to get the antiderivative: 2x⁴ + x³ + 3x².
- Apply the limits: compute F(b) - F(a), where F(x) is the antiderivative.
- Substitute b (upper limit) and a (lower limit) into F(x), then subtract the results.
Example Calculation
- Definite integral from x = 2 to x = 3 of 8x³ + 3x² + 6x:
- Antiderivative: 2x⁴ + x³ + 3x².
- F(3) = 2(3⁴) + 3³ + 3(3²) = 162 + 27 + 27 = 216.
- F(2) = 2(2⁴) + 2³ + 3(2²) = 32 + 8 + 12 = 52.
- Definite integral value: 216 - 52 = 164.
Interpretation
- The value of a definite integral represents the area under the curve of the function between the specified x-values.
Key Terms & Definitions
- Definite Integral — Integral with upper and lower limits, giving a specific value.
- Indefinite Integral — Integral without limits; solution includes a constant.
- Antiderivative — Function whose derivative is the original function (F(x) for f(x)).
Action Items / Next Steps
- Review additional example problems from the provided video description links.
- Practice evaluating definite integrals with more complex functions, such as those involving square roots.