7.6 Improper Integrals

Question 1

How do you evaluate an improper integral with an infinite limit in terms of a formula?

Answer 1

∫(a on the bottom, ∞ on the top) f(x)dx=lim t→∞∫(a on the bottom, t on the top) f(x)dx

Question 1

What is an improper integral?

Answer 1

An integral is improper if its limits of integration are infinite or if the integrand has a discontinuity within the interval of integration.

Question 2

How do you evaluate an improper integral with a discontinuity at a point
c in [a,b]?

Answer 2

Split the integral at the discontinuity and use limits:

∫(a on the bottom, b on the top) f(x)dx=lim t→c−∫ ∫(a on the bottom, t on the top) f(x)dx+lim t→c- ∫(b on the bottom, t on the top) f(x)dx

Question 3

What does it mean for an improper integral to converge?

Answer 3

if the limit(s) exist and are finite.

Question 4

What does it mean for an improper integral to diverge?

Answer 4

if the limit(s) do not exist or are infinite.

Question 5

What is the general method for evaluating improper integrals?

Answer 5

1. Rewrite the integral as a limit.
2. Evaluate the limit (if it exists) to determine convergence or divergence

Question 6

How do you determine if an improper integral converges or diverges?

Answer 6

Check the function’s behaviour as x→∞ (or near a discontinuity). If the integral tends to a finite number, it converges; otherwise, it diverges.

Question 7

How do you analyze the behavior of an improper integral with a discontinuity?

Answer 7

Approach the point of discontinuity from both sides using limits. If both limits exist and are finite, the integral converges.