Improper Integrals

A Problem With Riemann Integrals

Explanation that Riemann integrals are not defined when the interval is unbounded but can often be found using limits. Mention of the alternative Lebesgue integral.

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

Course Material Related to This Topic:

• Read lecture notes, section 1 on pages 1–2

Improper Integrals of the First Kind

Using limits to evaluate improper integrals with unbounded limits of integration. Includes examples of integrating 1/(x^p) from 1 to infinity and integrating cos(x) from 0 to infinity.

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

Course Material Related to This Topic:

• Read lecture notes, section 2 on pages 2–3

Improper Integrals of the Second Kind

Using limits to evaluate improper integrals involving functions that are unbounded over the specified limits of integration. Includes example of integrating 1/(x^p) between 0 and 1.

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

Course Material Related to This Topic:

• Read lecture notes, section 3 on pages 3–4

The Comparison Test

Definition of monotone bounded limits, the squeezing lemma for limits and improper integrals, and the comparison test for convergence of improper integrals.

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

Course Material Related to This Topic:

• Read lecture notes, section 4 on pages 4–5

Improper Integrals

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

Course Material Related to This Topic:

The Comparison Test for determining convergence or divergence of improper integrals, with discussion and examples.

• Read lecture notes, page 1 to page 3

An integral with an infinite upper limit of integration to be evaluated.

• Complete exam problem 16 on page 2
• Check solution to exam problem 16 on page 1

Determining whether twenty-two different improper integrals are convergent or divergent, and evaluating the limits of six integrals using the Fundamental Theorem.

• Complete exam problems 6B-1 to 6B-8 on page 40
• Check solution to exam problems 6B-1 to 6B-8 on pages 91–3

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

Course Material Related to This Topic:

Determining whether an improper integral converges or diverges.

• Complete exam problem 4 on page 1
• Check solution to exam problem 4 on page 2

Two questions which involve determining whether an improper integral will converge or diverge.

• Complete exam problems 8.1 to 8.2 on page 6
• Check solution to exam problems 8.1 to 8.2 on page 6