12.3.1 The First Type of Improper Integral Flashcards Preview

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Flashcards in 12.3.1 The First Type of Improper Integral Deck (10):
1

The First Type of Improper Integral

An improper integral is a definite integral with one of the following properties: the integration takes place over an infinite interval, or the integrand is undefined at a point within the interval of integration.
• With some improper integrals, the area of the region under the curve is finite even though the region extends to infinity.
• An improper integral that is infinite diverges. An improper integral that equals a numerical value converges.

2

note

- A definite integral is considered an improper integral if it has one of these properties:
· the integration is over an infinite interval; or
· its integrand is undefined at a point within the
interval of integration.
- Imagine calculating the area of the region under the curve y=1/x^2, starting at x = 1 and moving to the right. You would integrate the function from 1 to ∞ and solve the improper integral. The solution of the integral is one. This means that the area under the curve to the right of x = 1 has an area of one. You can think about this area “repackaged” into the square bounded by the origin and the point (1,1). Because this improper integral has a finite value, it converges.
- An improper integral diverges if its value is infinite. Imagine calculating the area under the curve y=1/x
, starting at x = 1 and moving to the right. You would integrate the function from 1 to and solve the improper integral. In this example, the value of the integral is infinity, meaning that the area under the curve is infinitely large. This improper integral diverges.

3

Evaluate ∫∞ 25 250,000/x^3dx.

200

4

Which of the following best describes the red region under the curve for y = 1 / x^ 2?

The red region is dropping in height as x increases. As a result, the change in the area decreases as x increases

5

Which of the following statements correctly describes improper integrals?

An improper integral that has a defined numerical value converges. Otherwise the improper integral diverges.

6

Evaluate ∫∞ 1 1/√x dx

The improper integral diverges.

7

Which of the following is not an improper integral?

∫−2 0 x dx

8

Which of the following statements is true for
∫ ∞ 0 e^−x dx?

The integral is improper and converges to 1.

9

Evaluate ∫ ∞ 0 cosx dx.

The improper integral diverges.

10

Which of the following best explains why these two improper integrals differ from one another?
∫∞11x2dx∫∞11xdxImproper Integral A Improper Integral B

Improper Integral A is convergent because it approaches a value of 1.
Improper Integral B is divergent because it approaches infinity.

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