Mastering Definite Integrals with FTC Part 2

May 5, 2024

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Summary of Lecture on Evaluating Definite Integrals

The lecture covered the evaluation of definite integrals, explaining the difference between indefinite and definite integrals, the integral sign, limits of integration, and the application of the Fundamental Theorem of Calculus Part 2 (FTC Part 2). Several examples were solved to demonstrate how to compute definite integrals from various functions.

Key Concepts and Definitions

  • Indefinite Integral vs. Definite Integral:

    • Indefinite Integral: Does not have upper and lower limits; it results in a function plus a constant of integration (C).
    • Definite Integral: Has upper (b) and lower (a) limits of integration, known as limits of integration; it results in a specific numerical value.

  • Integral Components:

    • Integrand (f(x)): The function to be integrated.
    • dx: Indicates the variable of integration.
    • Integral Sign (∫): Represents a limit of sums.

  • Fundamental Theorem of Calculus Part 2:

    • The definite integral of f from a to b equals F(b) - F(a), where F is the antiderivative of f.

Examples and Calculations

  1. Example 1:

    • Problem: Evaluate the definite integral of 8 dx from 2 to 5.
    • Steps:
      • Find the antiderivative of 8, which is 8x.
      • Evaluate ( F(5) - F(2) ):
        • ( F(5) = 8 \times 5 = 40 )
        • ( F(2) = 8 \times 2 = 16 )
        • ( 40 - 16 = 24 )
    • Result: 24

  2. Example 2:

    • Problem: Find the definite integral of ( 5x - 4 ) from 1 to 4.
    • Steps:
      • Use the power rule to find antiderivatives:
        • ( \int 5x = \frac{5x^2}{2} )
        • ( \int -4 = -4x )
      • Evaluate ( F(4) - F(1) ):
        • Perform the calculations to find out the values at ( F(4) ) and ( F(1) ).
    • Result: ( \frac{51}{2} ) or 25.5

  3. Example 3:

    • Problem: Evaluate integral of ( 8x^{-3} ) from -3 to 4.
    • Steps:
      • Rewrite ( x^{-3} ) properly to use the power rule, finding the antiderivative.
      • Simplify and evaluate ( F(4) - F(-3) ).
    • Result: ( \frac{7}{36} )

  4. Example 4:

    • Problem: Evaluate the definite integral from 1 to e of ( \frac{5}{x} ) dx.
    • Steps:
      • Recognize ( \int \frac{1}{x} ) dx as ( \ln(x) ).
      • Evaluate ( F(e) - F(1) ) using the properties of natural logarithms.
    • Result: 5

  5. Example 5:

    • Problem: Integral of the square root of x from 4 to 9.
    • Steps:
      • Convert ( \sqrt{x} ) to ( x^{1/2} ) and apply power rule.
      • Evaluate ( F(9) - F(4) ).
    • Result: 2

These examples demonstrate how to find the antiderivative, simplify expressions, and use the limits of integration to calculate the value of a definite integral effectively.