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Cálculo Integral: Integrales Impropias y Transformadas de Laplace, Apuntes de Cálculo

The regions we considered on Friday were infinite horizontally. • Today we’ll consider regions that are infinite vertically. • Example:

Tipo: Apuntes

2018/2019

Subido el 16/03/2023

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Math 1220-3
Notes of 2/27/23
Note that hw 8, which opened today, will close
on a Wednesday (March 15), not on a Friday.
The purpose of this early closure is to make
sure everybody finishes the hw before our next
exam on March 17.
8.4 More Improper Integrals
The regions we considered on Friday were in-
finite horizontally.
Today we’ll consider regions that are infinite
vertically.
Example:
A=!1
1
1
1x2dx
Figure 1 shows the graph of the integrand.
Math 1220-3 Notes of 2/27/23 page 1
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Math 1220-

Notes of 2/27/

  • Note that hw 8, which opened today, will close on a Wednesday (March 15), not on a Friday. The purpose of this early closure is to make sure everybody finishes the hw before our next exam on March 17.

8.4 More Improper Integrals

  • The regions we considered on Friday were in- finite horizontally.
  • Today we’ll consider regions that are infinite vertically.
  • Example:
A =

− 1

1 − x^2

dx

  • Figure 1 shows the graph of the integrand.

0

1

2

y

–1 –0.8 –0.6 –0.4 –0.2 0.2 0.4 0.6 0.8 1 x

Figure 1. Graph of integrand.

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Figure 2. Diamond.

i

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suppose

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Gabriel’s Horn

  • A well known paradox. Suppose the graph of y = (^) x^1 where x ≥ 0 is rotated around the x-axis. Compute its volume and its surface area. We’ll find that the volume is finite, but the area is infinite. How can that be?
  • As we discussed in Calc I, the formulas for the volume and the surface area of a solid of revolution (of the graph of f around the x- axis) are given by
V =

∫ (^) b

a

π

f (x)

dx

and

A =

∫ (^) b

a

2 πf (x)

f ′(x)

dx.

See Section 5.4 in the textbook for more in- formation.

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  • If time allows compute L {sin t}.

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