Evaluate Definite Integrals

The Fundamental Theorem of Calculus

• 18.01 Single Variable Calculus, Fall 2005
  Prof. Jason Starr

Course Material Related to This Topic:

Fundamental Theorem of Calculus is presented and justified.

• Read lecture notes, section 3 pages 5–6

• 18.013A Calculus with Applications, Spring 2005
  Prof. Daniel J. Kleitman

Course Material Related to This Topic:

Statement and explanation of the two forms for the Fundamental Theorem of Calculus.

• Read chapter 21 of online textbook

• 18.01 Single Variable Calculus, Fall 2006
  Prof. David Jerison

Course Material Related to This Topic:

Using the Fundamental Theorem to evaluate an integral.

• Complete exam problem 5 on page 1
• Check solution to exam problem 5 on page 1

Six questions which involve finding areas under curves and evaluating definite integrals.

• Complete exam problem 3C–1 on page 22 to problem 3CC–6 on page 22
• Check solution to exam problems on page 40

Algorithm for Computing Riemann Integrals

General algorithm for evaluating integrals using the Fundamental Theorem of Calculus.

• 18.01 Single Variable Calculus, Fall 2005
  Prof. Jason Starr

Course Material Related to This Topic:

• Read lecture notes, section 4 page 7

The Two Fundamental Theorems of Calculus

Statement and explanation of the First and Second Fundamental Theorems, with examples.

• 18.01 Single Variable Calculus, Fall 2006
  Prof. David Jerison

Course Material Related to This Topic:

• Read lecture notes, section 1 pages 1–2

Proof of the Two Fundamental Theorems

An intuitive argument justifying the Second Fundamental Theorem and a complete proof of the First Fundamental Theorem, including remarks about these theorems as they relate to definite and indefinite integrals.

• 18.01 Single Variable Calculus, Fall 2006
  Prof. David Jerison

Course Material Related to This Topic:

• Read lecture notes, section 4 pages 6–7

Functions Defined By Integrals

• 18.01 Single Variable Calculus, Fall 2006
  Prof. David Jerison

Course Material Related to This Topic:

Three part question finding bounds on the values of a function defined in terms of an integral and finding where the function is concave up or concave down.

• Complete exam problem 4 on page 1
• Check solution to exam problem 4 on page 1

Four part question which involves finding the derivative, critical points, and an estimation of a function which is defined in terms of an integral.

• Complete exam problem 3 on page 1
• Check solution to exam problem 3 on page 1

Four part question which involves finding critical points and values of a function which is defined in terms of an integral.

• Complete exam problem 5 on page 2
• Check solution to exam problem 5 on page 2

Finding the first and second derivatives of a function defined in terms of an integral, and expressing another integral in terms of that function.

• Complete exam problem 10 on page 2
• Check solution to exam problem 10 on page 1