Lecture Notes: Fundamental Theorem of Calculus

Overview

• Review of avoiding limits in evaluating definite integrals using the Fundamental Theorem of Calculus.
• Introduction and exploration of the Second Part of the Fundamental Theorem of Calculus.

Fundamental Theorem of Calculus, Part 2

• Applies to continuous functions on a closed interval [A, B].
• Theorem Statement:
  • If f(x) is continuous on [A, B] and F(x) is an antiderivative of f(x), then: [ \int_{A}^{B} f(x) , dx = F(B) - F(A) ]
  • Connects to the Riemann sum concept but avoids cumbersome limits.

Evaluating Definite Integrals

• Steps:
  1. Find the antiderivative F(x) of f(x).
  2. Evaluate F at the upper limit (B) and subtract evaluation at the lower limit (A).
  3. [ \text{Result: } F(B) - F(A) ]
• Notation: The constant of integration (+C) cancels out and is often ignored in definite integrals.

Examples of Evaluating Integrals

• Example 1:

  • [ \int_{0}^{2} (x^2 + 1) , dx ]
  • Antiderivative: [ F(x) = \frac{1}{3}x^3 + x ]
  • Evaluate: [ F(2) - F(0) = \frac{8}{3} + 2 - 0 = \text{14/3 square units} ]

• Example 2:

  • [ \int_{-1}^{3} (2 + x) , dx ]
  • Antiderivative: [ F(x) = 2x + \frac{1}{2}x^2 ]
  • Evaluate: [ F(3) - F(-1) = 12 ]

Simplifying Complex Integrals

• Bringing constants outside and simplifying before integrating.
• Example with natural logs showing importance of absolute value.

Special Cases

• Recognizing integrals that resolve to simple geometric shapes or known values (e.g., rectangle area).

Solving Integral Equations

• Use fundamental theorem to find value of limits for a given integral result.
• Example: Solving for x in [ \int_{0}^{x} (4t - 9) , dt = -4 ].
  • Antiderivative: [ F(t) = 2t^2 - 9t ]
  • Solve: [ 2x^2 - 9x = -4 ] leading to solutions for x.

Applications

• Useful for finding areas, solving integral equations, and evaluating integration bounds.
• Prepares for more complex integration techniques in advanced mathematics courses.

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This lecture provides foundational understanding of calculating definite integrals using the second part of the Fundamental Theorem of Calculus, bypassing complex limits and simplifying calculations through antiderivatives.