Need help with line integral question

[hollow temple]

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[idle mirage]

You know how to parameterize that line?

[clear vector]

i dont

i forgot actually

[valid umbra]

ok so you have the lone of gradient two and intercept 0, meaning you have y=2x and you set x=t

meaning you have y=2t and x=t

[idle mirage]

Does brodie know how a line int is defined with a parametrization?

[brave marsh]

apex predator

[valid umbra]

0≤t≤1

[brave marsh]

apex predator

[clear vector]

why the root tho

[valid umbra]

yknow Pythagoras?

[clear vector]

ye

[valid umbra]

yeah that's why, your gradient for your x and your y as defined by t (which is the gradient) is scaled by the distance between your gradients in x and y and 0

[brave marsh]

apex predator

[valid umbra]

seeing as it's a straight line the gradients are constant which means your line has the same scale factor across the entire line segment

[clear vector]

its just a 3 mark question why are there so many ga damn steps

[valid umbra]

see Professor Trefor Bazetts video on line integrals for a far more comprehensive and visual explanation

[valid umbra]

there aren't, really

[idle mirage]

Cuz you were probably taught it before lol? Apex is just filling in the gaps you have

Also I think your protege is getting confused @valid umbra check general

[clear vector]

we got y = 2x and x = t

if we just put it into the eqn and integrate wouldnt we get the answer?

https://www.youtube.com/watch?v=cPuuLjZN-to&ab_channel=MichelvanBiezen like this perhaps

[valid umbra]

no, you have to scale the line segment

because it's not running directly from 0 to t along the number line

it's travelling diagonally, by √5 units for every 1 horizontally

makes sense?

[clear vector]

not yet

could u explain how we get x and y from (0,0) and (1,2) like im a 5yr old

[idle mirage]

Do you remember how to write an equation for a line between two points?

[clear vector]

by finding the slope?

[valid umbra]

yes

(2-0)/(1-0)

@clear vector

[clear vector]

wouldnt that be 2

[valid umbra]

yes

so the gradient of your line is 2

what is it's y intercept?

[clear vector]

wdym

if its travelling from the line 0 to t

then t i guess

[valid umbra]

no, y intercept is what is the value of y when x=0

you're given the point (0,0)

which means y=? when x=0

[clear vector]

0

[valid umbra]

yes, so what is your y in terms of x

[clear vector]

how tf do i not get it

[valid umbra]

what's the equation for a line

equation

[clear vector]

i said 0 cuz(x,y) (0,0)

[valid umbra]

yes, and that's correct

what is the equation for a line

just a plane old straight line

[clear vector]

mx+c?

[valid umbra]

equation, not expression

what does mx+c equal

[clear vector]

y

[valid umbra]

yes, y=mx+c

you've worked out your m and your c

what is your y in terms of x

[clear vector]

it would be y-c\m

[valid umbra]

no

[clear vector]

oh wait mb

dont get it

[valid umbra]

y=mx+c, m is your gradient, c is your y intercept

what have you calculated your gradient and your y intercept to be

[clear vector]

so we just need to input the value then

[valid umbra]

yes, to find your function for y in terms of x

[clear vector]

y intercept is 0 and gradient is 2

[valid umbra]

yes, now put that in your equation

[clear vector]

y =2(x)

[valid umbra]

yes, now we can let x=t because that allows us to get the simplest form for y in terms of t

if x=t then y=?

[clear vector]

2t

[valid umbra]

yes

x=t
y=2t

[clear vector]

i got my sem final on monday how am i gonna pass that shit oof

[valid umbra]

now this

[clear vector]

what was this fromula for again?

[valid umbra]

the scaling factor

wait first you should probably find the bounds for t

if x=t, what bounds of t get 0≤x≤1

[clear vector]

0<t<1

[valid umbra]

yes

now find the scale factor

[clear vector]

i got sqrt2t

[sterile gyro]

@valid umbra is this like the one where if the graph was a line and we stretch it out its length is what we need

[valid umbra]

no

[valid umbra]

yes

[valid umbra]

what's dy/dt of 2t

[sterile gyro]

and what is he struggling with?

[valid umbra]

and what's dx/dt of t

[valid umbra]

the whole parameterising process

[clear vector]

im getting confused between derivation and integration

[sterile gyro]

lemme simplify this for him wait

just a second sir!

[valid umbra]

d/dt of t is 1

which means d/dt of 2t is?

@clear vector

[clear vector]

2

[valid umbra]

yes, now you plug that in for your scale factor

and get ?

[clear vector]

sqrt2

[valid umbra]

no

look at the formula again

[valid umbra]

this

[clear vector]

what is 0/dt

[valid umbra]

you have your dx/dt and your dy/dt

what

[clear vector]

oh wait nvm

it would be 3

[valid umbra]

not quite

you see that the derivatives are squared

[clear vector]

the the the root will cancel it na?

[valid umbra]

no, the square root operation is not linear

it doesn't cancel the addition of squares

[clear vector]

it would be sqrt5 then

[sterile gyro]

don't take my word for it

this is if I didn't butcher it

[valid umbra]

yes that's correct

[valid umbra]

never line integrate again

[sterile gyro]

👹

[clear vector]

and then?

[valid umbra]

yes that would be correct

you have the bounds for t, you have your scale factor, you have your x in terms of t and your y in terms of t

now you just need to substitute them all in

[clear vector]

where do i put in the scale factor tho

[valid umbra]

you just multiply the whole thing by it

it's scaling the whole integral yk

[clear vector]

so integral o to 1 sqrt5{3[(2t)(t)]}

[brave marsh]

apex predator

[clear vector]

its dy

yes

[valid umbra]

no, look at my formula

your dy=that scale factor dt

[valid umbra]

now you have it in that form you integrate as usual

2√5 is your end integral

you should watch Dr Trefor Bazett's videos on line integrals

they'll help you very much on your semis

[clear vector]

well there is only one day left

excluding today

[valid umbra]

https://youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa

the first 4 videos

[clear vector]

i know all the questions that will be asked i just dont know how to solve em uff

[sterile gyro]

kIllER! do you have tensor calculus by any chance?

[valid umbra]

bro

[clear vector]

tensor calc?

probably not

[clear vector]

hey can i slide in one more question?

[valid umbra]

I gtg, I'm sorry

ask it in a separate thread, someone else will help you with it

[clear vector]

alr close this one thanks

[valid umbra]

+close