- 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 is continuous on , then there exists at least one point such that:
- Geometric Interpretation:
- equals the average value of the function over
- The area of the rectangle with base and height equals the area under the curve
- Alternative Form:
- 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 over the interval and find such that equals the average value of the function over
Given find such that equals the average value of over
Fundamental Theorem of Calculus Part 1: Integrals and Antiderivatives
The Main Idea
- Theorem Statement:
- If is continuous on , and , then for all
- 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 :
- is a function that gives the area under the curve of from to
- The rate of change of this area function is the original function
- Connection to Anti-differentiation:
- is an antiderivative of
- All antiderivatives of 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
Let Find
Let Find
Fundamental Theorem of Calculus, Part 2: The Evaluation Theorem
The Main Idea
- Theorem Statement:
- If is continuous on and is any antiderivative of , then:
- 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:
- is often written as
- 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:
Suppose James and Kathy have a rematch, but this time the official stops the contest after only sec. Does this change the outcome?