12.3.1 The First Type of Improper Integral

Question 1

The First Type of Improper Integral

Answer 1

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.

Question 2

note

Answer 2

• 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.

Question 3

Evaluate ∫∞ 25 250,000/x^3dx.

Answer 3

200

Question 4

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

Answer 4

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

Question 5

Which of the following statements correctly describes improper integrals?

Answer 5

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

Question 6

Evaluate ∫∞ 1 1/√x dx

Answer 6

The improper integral diverges.

Question 7

Which of the following is not an improper integral?

Answer 7

∫−2 0 x dx

Question 8

Which of the following statements is true for

∫ ∞ 0 e^−x dx?

Answer 8

The integral is improper and converges to 1.

Question 9

Evaluate ∫ ∞ 0 cosx dx.

Answer 9

The improper integral diverges.

Question 10

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

Answer 10

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