12.3.3 Infinite Limits of Integration, Convergence, and Divergence

Question 1

Infinite Limits of Integration, Convergence, and Divergence

Answer 1

• Improper integrals can be expressed as the limit of a proper integral as some parameter approaches either infinity or a discontinuity

Question 2

note

Answer 2

• Formalizing the idea of improper integrals involves
  replacing the infinite endpoint with a parameter whose limit approaches either infinity or the discontinuity.
• There are three types of improper integrals over an infinite interval:
• In the first integral to the left, the right endpoint is infinite. To formalize this integral is replaced with b and the integral is evaluated as .
• In the second integral, the left endpoint approaches negative infinity. To formalize this integral – is replaced with a and the integral is evaluated as .
• In the third integral, the range of integration is the entire x-axis. Split the integral into the sum of two integrals each of which has a limit of integration at some midpoint, t. The first integral can be evaluated as in example 2 above and the second can be evaluated as in example 1.
• In this case the integral is improper because its domain has a discontinuity. Split the integral into the sum of two integrals each of which has a limit of integration at the discontinuity, x = c. The first integral is formalized by replacing c with E and evaluating the integral as . (Note that means E approaches c from the left or negative side of the x-axis.) The second integral is formalized by replacing c with D and evaluating the integral as . ( indicates that D approaches c from the right or positive side of the x-axis.)

Question 3

Which of the following expressions is equivalent to the improper integral
∫∞af(x)dx?

Answer 3

limb→∞∫baf(x)dx

Question 4

To evaluate the improper integral∫5−51x2dx,at which values of x should you break the integral?

Answer 4

x = 0

Question 5

Evaluate ∫2−2 dx/x^4

Answer 5

The integral diverges.

Question 6

Consider the red region under the curve y=f(x) where x→∞. Which of the following expressions correctly describes the limit of the area of the red region?

Answer 6

limb→∞∫baf(x)dx

Question 7

Evaluate ∫∞−∞−11+x2dx

Answer 7

-π

Question 8

Evaluate∫1−11√1−x2dx.

Answer 8

π

Question 9

Consider the red region under the curvey=f(x) where x→∞. Which of the following statements about the area isnot correct?

Answer 9

• The area is equal to the improper integral
  ∫∞af(x)dx.
• If the value of the improper integral is finite, then the integral converges.
• If the value of the improper integral is infinite, then the integral diverges.

Question 10

To evaluate the improper integral∫π0sec2xdx,at which values of x should you break the integral?

Answer 10

π / 2

Question 11

Evaluate ∫∞0e−xdx.

Answer 11

1