The Fundamental Theorem of Calculus: Fresh Take

  • Understand the Mean Value Theorem for Integrals and both components of the Fundamental Theorem of Calculus
  • Use the Fundamental Theorem of Calculus to find derivatives of integral functions and calculate definite integrals
  • Describe how differentiation and integration are interconnected

The Mean Value Theorem for Integrals

The Main Idea 

  • Theorem Statement:
    • If f(x) is continuous on [a,b], then there exists at least one point c[a,b] such that: f(c)=1baabf(x)dx
  • Geometric Interpretation:
    • f(c) equals the average value of the function over [a,b]
    • The area of the rectangle with base ba and height f(c) equals the area under the curve
  • Alternative Form:
    • abf(x)dx=f(c)(ba)
  • Significance:
    • Guarantees the existence of a point where the function takes on its average value
    • Bridges discrete and continuous concepts of average
  • The Mean Value Theorem for Integrals provides the theoretical basis for the average value of a function

Find the average value of the function f(x)=x2 over the interval [0,6] and find c such that f(c) equals the average value of the function over [0,6].

Given 03(2x21)dx=15, find c such that f(c) equals the average value of f(x)=2x21 over [0,3].

Fundamental Theorem of Calculus Part 1: Integrals and Antiderivatives

The Main Idea 

  • Theorem Statement:
    • If f(x) is continuous on [a,b], and F(x)=axf(t)dt, then F(x)=f(x) for all x[a,b]
  • Key Implications:
    • Integration and differentiation are inverse processes
    • Every continuous function has an antiderivative
    • Provides a way to evaluate definite integrals without using Riemann sums
  • Interpretation of F(x):
    • F(x) is a function that gives the area under the curve of f(t) from a to x
    • The rate of change of this area function is the original function f(x)
  • Connection to Anti-differentiation:
    • F(x) is an antiderivative of f(x)
    • All antiderivatives of f(x) differ by a constant
  • Applications:
    • Simplifies the process of finding antiderivatives
    • Allows for the evaluation of complex definite integrals
    • Forms the basis for many advanced calculus techniques

Use the Fundamental Theorem of Calculus, Part 1 to find the derivative of g(r)=0rx2+4dx.

Let F(x)=1x3costdt. Find F(x).

Let F(x)=xx2costdt. Find F(x).

Fundamental Theorem of Calculus, Part 2: The Evaluation Theorem

The Main Idea 

  • Theorem Statement:
    • If f is continuous on [a,b] and F(x) is any antiderivative of f(x), then: abf(x)dx=F(b)F(a)
  • Key Implications:
    • Provides a simple method to evaluate definite integrals
    • Connects the concept of area under a curve to the difference of antiderivative values
    • Eliminates the need for limits of Riemann sums in many cases
  • Notation:
    • F(b)F(a) is often written as F(x)|ab
  • Important Points:
    • Any antiderivative can be used (constant of integration cancels out)
    • The theorem applies to continuous functions on a closed interval
    • The result can be positive, negative, or zero, depending on the function and interval
  • Applications:
    • Simplifies calculations in physics, engineering, and economics
    • Allows for quick evaluation of accumulated change over an interval

Evaluate the following integral using the Fundamental Theorem of Calculus, Part 2:

19x1xdx.

Suppose James and Kathy have a rematch, but this time the official stops the contest after only 3 sec. Does this change the outcome?