calculu assignmnet
Angela gomez
cal2.pdf
JZfa20 Math 1920 Name: ___________________
Improper Integrals
Show all pertinent work to receive credit. Label and scale the axes of all your graphs.
An improper integral involves the computation of an unbounded area.
- integrals involving a horizontally unbounded region:
A f x dx f x dx a
b a
b
( ) lim ( )
- integrals involving a vertically unbounded region:
A f x dx f x dx b
b
( ) lim ( ) 0
1
0
1
These integrals may converge to a finite value, else they are said to diverge:
EXAMPLE 1. A horizontally unbounded region that diverges:
A x
dx x
dx x x b
b
b x x b
b
1 1
1 1 1lim lim ln lim ln
whereas
A horizontally unbounded region that converges:
~A x
dx x
dx x bb
b
b x
x b
b
1 1 1 1
1 1 0 12
1 2
1 1
lim lim lim
EXAMPLE 2. A vertically unbounded region that diverges:
A x
dx x
dx x bb b b x b
x
b
1 1 1 1 12
0
1
0 2
1
0
1
0 lim lim lim
whereas
A vertically unbounded region that converges:
~ A x
dx x
dx x b b
b b x b
x
b
1 1 2 2 1 2
0
1
0
1
0
1
0 lim lim lim ( )
EXERCISES:
1. Express the improper integral as a limit and compute its exact value:
a. = 1 2
1 x dx
Conclusion: The integral diverges / converges to _____________.
b. = 1
1 x dx
Conclusion: The integral diverges / converges to _____________. c. Complete the statement:
The integral converges for n values __________ , diverges for n values ____________. 1
1 x dxn
2. Express the improper integral as a limit and compute its exact value:
a. = 1 2
0
1
x dx
Conclusion: The integral diverges / converges to _____________.
b. = 1
0
1
x dx
Conclusion: The integral diverges / converges to _____________. c. Complete the statement:
The integral converges for n values __________ , diverges for n values ____________. 1
0
1
x dxn
3. Express the improper integral as a limit and compute the exact value of the improper integral
a. xe dxx
0
____________________. xe dxx
0
b. xe dxx
2
0
____________________.xe dxx
2
0
4. Express the improper integral as a limit and compute the exact value of the improper integral. A tabular scheme is recommended.
x e dxx
2 2 2
____________________.xe dxx
2
0
5. Express the improper integral as a limit and compute its exact value:
a. = 1
30
3
x dx
Conclusion: The integral diverges / converges to _____________.
b. =
1 3 20
3
x dx
Conclusion: The integral diverges / converges to _____________.
6. Consider the horizontally unbounded area A represented by the integral:
, where a is a positive constant. A x e dx ax
( )1 2 0
a. Express the improper integral as a limit and compute the value of the improper integral
7. Consider the horizontally unbounded area A represented by the integral:
, where a is a positive constant. A x a e dx x
( )2 0
a. Express the improper integral as a limit and compute the value of the improper integral
8. Express the improper integral as a limit and compute the value of the integral. The constant a is positive.
a. = 1
0 a x dx
a
Conclusion: The integral diverges / converges to _____________.
b. = 1
2 0 ( )a x
dx a
Conclusion: The integral diverges / converges to _____________.
b. =
1
0 a x dx
a
Conclusion: The integral diverges / converges to _____________.
9. Express the improper integral as a limit and compute the value of the integral. The constant a is positive.
a. = 1
2 2 0 a x
dx a
Conclusion: The integral diverges / converges to _____________.
b. = 1
2 2 0 a x
dx a
Conclusion: The integral diverges / converges to _____________.
c. = x
a x dx
a
2 2 0
Conclusion: The integral diverges / converges to _____________.
d. = x
a x dx
a
2 2 0
Conclusion: The integral diverges / converges to _____________.
