6.8 Improper Integrals

Question 1

what does convergent mean

Answer 1

finite answer, exists

Question 2

what does divergent mean

Answer 2

infinite answer, kinda does not exist

Question 3

imagine the graphs of
x
1/x
x^2
square root of x
a^x
ln x
sin x
cos x
tan x

Answer 3

hehe

Question 4

how to rewrite the improper integral from a to infinity of f of x

Answer 4

lim as t approaches infinity of the integral from a to t of f of x

Question 5

how to rewrite the integral from negative infinity to b of f of x

Answer 5

lim as t approaches negative infinity of the integral from t to b of f of x

Question 6

how to rewrite the integral from negative infinity to positive infinity

Answer 6

integral from negative infinity to a plus the integral from a to infinity

then you can rewrite them with the lim but just one since if the first one is convergent or divergent, the second one is the same

Question 7

Simple steps to solve improper integrals with infinite bounds

Answer 7

1. rewrite with the limit
2. find antiderivative and evaluate at the bounds
3. pass the limit
4. figure out its behavior
5. answer or divergent

Question 8

what are the two types of improper integrals and their differences

Answer 8

over infinite bounds and finite bounds

both are improper just for different reasons
infinite - bounds are infinite
finite - vertical asymptote

Question 9

rewrite an improper integral from a to b with a vertical asymptote c

Answer 9

lim as t approaches vertical asymptote (c) from left of integral from a to t
plus
lim as s approaches vertical asymptote(c) from the right of integral from s to b

Question 10

simple steps for solving a finite improper integral - meaning bounds are finite but there is something weird aka a vertical asymptote at a bound

Answer 10

1. add the limit based on where the vertical asymptote is AND coming from the right direction
2. find antiderivative and evaluate at the bounds
3. pass the lim (CAREFUL WITH THE DIRECTION OF THE LIMIT)
4. find how it behaves
5. answer or divergent

Question 11

when do you use p-integrals

Answer 11

when you cannot integrate
OR
they JUST ask you if it is convergent or divergent

Question 12

what are the two p-integrals

Answer 12

integral from 1 to infinity of 1/ x^p
converges for p > 1
diverges for p <= 1

integral from 0 to 1 of 1/x^p
converges for p < 1
diverges for p >= 1

Question 13

when to use the comparison tests

Answer 13

when we cannot integrate the integral we came up with
we just need to know if it and can find if it converges or diverges

Question 14

what is the comparison test and simple steps

Answer 14

both positive function with f being less than g for all of x
if g converges, f converges
if f diverges, g diverges

1. make sure bound 1 is greater than 1 or 1
2. make sure f (the integrand) is greater than 0
3. simplify f and g into an equality
   - g is one we are making to compare to f (given)
   -many ways you can do it
   f >= or<= g
   - start with f as the most complicated thing in integrand and then modify both side to try to fit og function
   - 4 > 3 so 1/4 < 1/3
4. compare like test says

• in explanation
• the g you came up with diverges/converges bc p integral
• since inequality
• then given integral diverges/converges by comparison test

Question 15

what is the limit comparison test and simple steps

Answer 15

f and g are positive continuous function on a to infinity
if lim as x approaches infinity of f / g = a number L greater than 0
then f and g both converge or diverge

1. f is the integrand
2. find g
   - remove things that are small or not doing anything
   - -g should be convergent
3. divide f by g
4. pass lim
   it
5. IF you get a number use the test
6. if g diverges or converges, so does f

–in explanation
- got a number L between 0 and infinity then integral from 1 to 0 of f and g both converge or diverge by limit comparison test
- we know g diverges/converges bc p integral
- conclusion f also diverges/converges

Question 16

how should you look at the fractions to see what happens/behave (improper integrals) or which one is larger (comparison tests)

Answer 16

pizzas / people