Question

Need help with integration and differentiation!

Answer 1

I wish I could help.. Unfortunately I'm not that far

Answer 2

wait.. I may be able to help

Answer 3

Yeah i can. I did that last semester. What can I do to help??

Answer 4

If f is continuous for all x, determine which of the following integrals must have the same value.
\[1. \int\limits_{a}^{b}f(x)dx\]
\[2. \int\limits_{0}^{a+b}f(x+a)dx\]
\[3. \int\limits_{a-c}^{b-c}f(x+c)dx\]

Answer 5

Any of you four want to help me?

Answer 6

Dang... Not what I thought it was. I'm sorry I couldn't help.. I tried, but I don't even understand it.

Answer 7

I' a sophomore in highschool. I really wish I could help.. Here try this: https://mathway.com/

Answer 8

@freckles

Answer 9

try doing a substitution on the last two
you know we are trying to see if we can write the last two in the form of the first

Answer 10

do you understand what I'm saying?

Answer 11

like u-sub or something lik a=1

Answer 12

example for b
do a substitution u=x+a
and see if the upper and lower limits end up being the same as the first

Answer 13

do something similar for example c

Answer 14

I mean letter c

Answer 15

Okay so u= x+a

Answer 16

but they dont have any constants as the upper and lower end limits

Answer 17

are you saying to make the constants?

Answer 18

so you didn't mean to write
\[\int\limits_0^{a+b} f(x+a) dx ?\]
as the expression for b?

Answer 19

Yah that is the second expression

Answer 20

you meant to write
\[\int\limits f(x+a) dx ?\]

Answer 21

I thought you said you had no limits

Answer 22

if u=x+a
and x=a+b then u=?
and x=0 then u=?

Answer 23

call it expression 2 so i dont get confused about the variables in the limits

Answer 24

0??

Answer 25

or a

Answer 26

where do you get 0?

Answer 27

if x=0 then u=x+a=0+a=a yes
what about the upper limit

Answer 28

x

Answer 29

I do not know how you are getting that if u=x+a and if x=a+b then u=x?

Answer 30

if u=x+a and x=a+b then u=(a+b)+a=2a+b`

Answer 31

the given upper limit is a+b and you said x=a+b so upper limit = x

Answer 32

no

Answer 33

remember we made the sub u=x+a?

Answer 34

the upper limit is x=a+b
so u=x+a=(a+b)+a=2a+b

Answer 35

the new upper limit is therefore 2a+b not x

Answer 36

oh okay that makes sense i didnt know that we were still using the usub

Answer 37

that is the only way I know to do the problem is the sub I mentioned awhile back...
\[\int\limits\limits_{0}^{a+b}f(x+a)dx \\ \int\limits_{x=0}^{x=a+b} f(x+a) dx \\ u=x+a \\ \text{ if } x=a+b \text{ then } u=(a+b)+a =2a+b \\ \text{ if } x=0 \text{ then } u=0+a=a\]

Answer 38

so what does the integral look like now after the substitution

Answer 39

\[\int\limits_{0}^{2a+b}f(x+a)dx\]

Answer 40

not exactly

Answer 41

you have only wrote the limits in terms of u
you also need to write the whole thing in terms of u not just the limits

Answer 42

remember the x+a can be replaced with u since u=x+a
and dx=?

Answer 43

f(u) instead

Answer 44

dx= 1

Answer 45

took the derivative of x+a

Answer 46

and oh yeah not sure why you changed are earlier limit from u=0 to u=a for the lower limit
\[\text{ so do you think } \int\limits_a^b f(x) dx \text{ is the same as } \int\limits_a^{2a+b}f(u) du\]

Answer 47

dx actually equals du

Answer 48

since u=x+a

Answer 49

okay

Answer 50

are you thinking about my question?

Answer 51

no i dont think they are the same

Answer 52

if the different variables confuse you can you replace x with u in that first integral...
I will ask the same question again
\[\text{ do you think } \int\limits_a^b f(u) du \text{ is the same as } \int\limits_a^{2a+b} f(u)du\]

this may be equal for some functions f but not all since b doesn't equal 2a+b

Answer 53

I should say may be equal for some functions f with some conditions on b and a

but yep this is definitely not an identity for all functions f

Answer 54

you try the last integral

Answer 55

well yes then

Answer 56

yes to what?

Answer 57

the first two equal each other

Answer 58

? we already concluded that they are not

Answer 59

what changed your mind

Answer 60

sorry nevermind

Answer 61

what changed my ind is you said that you were going to ask again

Answer 62

ok I didn't think you were able to answer my question because there was a long pause

Answer 63

okay so for the third one u = x+c

Answer 64

x= a-c for the lower limit and x=b-c for the higher limit

Answer 65

ok those are the limits before the substitution

Answer 66

but after you make u=x+c
then the lower limit is?
and the upper limit is?

Answer 67

u=a because the c's cancel eachother out

Answer 68

-c and +c

Answer 69

have you figured out the upper limit yet?

Answer 70

and the upper limit is u=b

Answer 71

so what does the last integral look like after the sub you chose

Answer 72

\[\int\limits_{a}^{b}f(u)\]

Answer 73

du

Answer 74

which is the same as which question 1 or 2?

Answer 75

so they are all three the same

Answer 76

:p no

Answer 77

lol we already said the second one and the first one aren't the same

Answer 78

1,3

Answer 79

great

Answer 80

sorry

Answer 81

it's k
too many expressions :p

Answer 82

i got excited

Answer 83

this really helped me understand u subsitution more thank you!!

Answer 84

the 1st one and 2nd one can be same under certain conditions
but it is not true for all f or a or/and b
but the point is they aren't the same for all f and a and/pr b

Answer 85

okay thank you

Answer 86

there is one more type of problem that i dont understand derivatives of definite integrals

Answer 87

but yep you are right about 1st and 3rd being the same
those are the only ones that are the same in this question

just saying this just to scare away any confusion

Answer 88

ok

Answer 89

what is it

Answer 90

want me to post it on a different question or just here?

Answer 91

post as a new question just in case of unforseen future disturbs us