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:**

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:**

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:**

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:**

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.

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