12.3.3 Infinite Limits of Integration, Convergence, and Divergence Flashcards Preview

AP Calculus AB > 12.3.3 Infinite Limits of Integration, Convergence, and Divergence > Flashcards

Flashcards in 12.3.3 Infinite Limits of Integration, Convergence, and Divergence Deck (11):
1

Infinite Limits of Integration, Convergence, and Divergence

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

2

note

- 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.)

3

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

limb→∞∫baf(x)dx

4

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

x = 0

5

Evaluate ∫2−2 dx/x^4

The integral diverges.

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?

limb→∞∫baf(x)dx

7

Evaluate ∫∞−∞−11+x2dx

8

Evaluate∫1−11√1−x2dx.

π

9

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

- 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.

10

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

π / 2

11

Evaluate ∫∞0e−xdx.

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