Calc Question Help

[dim sky]

Can someone give some insight on how to complete this problem?

[hot mirage]

try changing the variable or simply u substitution

for first one take u=t/2

whats du/dt?

[dim sky]

isnt it just 1/2

[hot mirage]

yup

now..

du/dt=1/2
dt/du=2
dt=2du

now change the limits

how do you do that?

[dim sky]

is it 2du from 18 to 0

[hot mirage]

no no keep dt=2du aside

u=t/2

if t=0(present lower limit)
u=0 (limit after changing variable)

ryt?

[dim sky]

yeah

[hot mirage]

similarly if t=18
u=9

now whats our integral

[dim sky]

um

[hot mirage]

$$\int_0^{18} g(t/2)dt$$
$$\int_0^{9} g(u) 2du$$
$$2\int_0^{9} g(u) du$$

ryt?

[bitter bison]

pratham

[dim sky]

O

yeah

so would it just be 2 times 7 basically?

[hot mirage]

yup

similarly try second one

[dim sky]

do i split apart the second one?

[hot mirage]

no

just u=9-t

you cannot split it

its inside the function

[dim sky]

if u = 9-t

then du = -1?

[hot mirage]

du/dt is -1

not just du

alr what do you do next?

[dim sky]

do i make the function negative because du/dt is -1

then

[hot mirage]

umm?

dt=-du

[dim sky]

yeah

and then do i try to isolate the variable t?

using u

[hot mirage]

umm there is no need

wait i am facing a small problem

that is ..

u=9-t

[dim sky]

yeah

[hot mirage]

if t=0
u=9
if t=9
u=0

i got lower boundary 9 and upper 0

[dim sky]

O

yeah

[hot mirage]

should we just interchange them?

[dim sky]

cant we just flip it using a negative sign

if the upper bound is 0 and the lower bound is 9

[hot mirage]

umm lemme see

i will just take an example to see it that should be true that we can just flip the boundaries

[dim sky]

im like 99 percent sure you can flip it using a negative siugn

[hot mirage]

right

yes

[dim sky]

but what would U be

[hot mirage]

you flip it using negative sign

[hot mirage]

u=9-t ryt?

yes this is true

[dim sky]

then how do i find the answer using g(t)dt = 7

if it is - 9 to 0 g(9-t)dt

[hot mirage]

see its simple...

[dim sky]

i feel it im close to figuring it out

[hot mirage]

$$\int _0^9 g(9-t)dt$$
put u=9-t
$$\frac{du}{dt}=-1$$
$$dt=-du$$

for boundaries
if$$t=0\implies u=9$$
if$$t=9 \implies u=0$$

$$\implies \int_9^0 g(u)(-du)$$
$$-\int _9^0 g(u)du$$
$$\int_0^9 g(u)du$$

in fact this is a rememberable indentity used for definite integrals

$$\int_0^a f(x)dx = \int_0^a f(a-x)dx$$

[bitter bison]

pratham

[dim sky]

isnt u just 9-x

im confused

[hot mirage]

yup u=9-x

why confused?

[dim sky]

how do i use that to find the answer is 9-t doesnt equal a value

[bitter bison]

pratham

[hot mirage]

ok a small error sry

😅

[dim sky]

Im at this part but I dont get how to find the answer from g(9-t)du

[hot mirage]

g(9-t)dt

see if you are substituting some variable you are also supposed to change the variable with which you are integrating with respect to

that is the reason i was differentiating

was that your doubt or....

[dim sky]

My main problem is figuring out how to use g(t)dt = 7 in g(9-t)dt

[hot mirage]

so.. g(t)dt isnt 7
its the integral of g(t)dt from 0 to 9

[dim sky]

@hot mirage do you mind if we just call for like 1 sec so you can explain?

[hot mirage]

in definite integration..
$$\int_a^b f(u)du=\int_a^b f(v)dv$$

[bitter bison]

pratham

[hot mirage]

ummm...

no sry tere are quite disturbances here

[hot mirage]

in definite integrals variable doesnt matter only boundaries

was that your doubt?

[dim sky]

not really

so

basically what I have is g(9-t)dt

from 9 to 0

[hot mirage]

integral of..

yes

[dim sky]

but if I go from 9 to 0 isnt it just g(0) - g(9)

?

[hot mirage]

not g(0)-g(9)

its G(0)-G(9)

the anti derivative of g

[dim sky]

what does that equate too

because I can't input G(0) - G(9) into the answer

[hot mirage]

you dont have a function so you cannot find he anti derivative

[dim sky]

could I just see what your answer is?

because I feel like im just going in a circle

[hot mirage]

the solution..

[dim sky]

i know thats the formula

and how to do it

but I dont know what it equals

[hot mirage]

oh it equals 7 simply

[dim sky]

wait

why would it equal 7

[hot mirage]

coz its given lol

[hot mirage]

..

[dim sky]

its given 7

idk

ill ask my teacher

it makes sense but it doesnt make sense at the same time

[hot mirage]

lol

have you done such problems earlier?

like u substitution

[dim sky]

today is my first day on u subsistuation

and for the most part I get it