Fundamental Theorem of Calculus Study Notes

Fundamental Theorem of Calculus

Introduction to Functions and Intervals

• Let’s consider a function f that is continuous over the interval [c, d].

• The variables c and d are designated as endpoints of the interval; a and b will be used later.

Definition of Capital F

• We define a new function, capital F(x), as the area under the curve of function f from c to x (where x is within the interval).

• Mathematically, this can be represented as: F(x) = \int_{c}^{x} f(t) \, dt

  • This equation represents the area under the curve between points c and x.

Fundamental Theorem of Calculus - Part 1

• According to the Fundamental Theorem of Calculus:

  • If f is continuous over the interval [c, d], then F is differentiable at every point x in [c, d].

  • The derivative of F at point x is equal to f at that same point:

  • Expressed formally,
    F'(x) = f(x)

  • This relationship holds true for every x in the interval [c, d].

Introduction to Second Fundamental Theorem of Calculus

• The goal is to connect the first part of the Fundamental Theorem to the second part, which is used for evaluating definite integrals.

• Let’s analyze the expression of the areas under curve at the endpoints a and b, where a and b are also contained in the interval [c, d], and b is assumed to be larger than a.

Understanding Areas Under the Curve

• Define capital F(b) as the definite integral from c to b:

  • F(b) = \int_{c}^{b} f(t) \, dt

  • This represents the total area under the curve f from c to b.

• Define capital F(a) as the definite integral from c to a:

  • F(a) = \int_{c}^{a} f(t) \, dt

  • This represents the total area under the curve f from c to a.

Visualizing the Areas

• The area represented by F(b) is depicted as a blue area on a graphical representation.

• The area represented by F(a) is depicted as a magenta area on the same graph.

• When you subtract the area represented by F(a) from F(b):

  • F(b) - F(a)

  • You end up with the area indicated in green, which represents the area under the curve f between points a and b.

• This can be expressed mathematically as:

  • \int_{a}^{b} f(t) \, dt

Second Fundamental Theorem of Calculus

• This gives us a significant result, stating:

  • If f is continuous on the interval [a, b], then:

  • The definite integral of f from a to b is equal to the antiderivative F evaluated at b and a. Formally:

  • \int_{a}^{b} f(t) \, dt = F(b) - F(a)

• Thus, it defines capital F as an antiderivative of f:

  • This means that F is the antiderivative of f, and the theorem facilitates the evaluation of definite integrals through antiderivatives.

Conclusion

• In conclusion, the second part of the Fundamental Theorem of Calculus is central to integral calculus as it provides a method for evaluating definite integrals via antiderivatives.

• Normally, the representation is organized as:

  • \int_{a}^{b} f(t) \, dt = F(b) - F(a)

  • This foundational principle is crucial for understanding and applying integral calculus effectively.