Question

find the arc length of a curve

Answer 1

\[r(t)=tcosti+tsintj+\frac{ 2\sqrt{2} }{ 3}t ^{3/2}k\]

Answer 2

r(t) such that t>0 and t<pi

Answer 3

\[\large L=\int\limits_{a}^{b}\sqrt{r\prime(t)}dt\] just compute the derivative and sub un the integral

Answer 4

correction:
\[\large L= \int\limits_{a}^{b}\sqrt{\left| r\prime(t) \right|}dt\]
compute the derivative and sub in the integral

Answer 5

-sinti+costj

Answer 6

how do i do the other derivation

Answer 7

isn't it

\[\large L= \int\limits_{a}^{b}|r\prime(t)|dt\]

Answer 8

it is

Answer 9

because it is the magnitude of r(t) where the derivative is squared and square rooted

Answer 10

i don't know how about going next

Answer 11

find the derivative of your 3 functions

Answer 12

i get -sin(t)+cos(t)+sqrt(2t)

Answer 13

but i dont think its right

Answer 14

you get that for what exactly

Answer 15

as is typical for these problems, the functions above are chosen just right so that the integration is trivial. if you end up with a difficult integral then you did something wrong

Answer 16

\[\sqrt{(-sint)^2+(cost)^2+(\sqrt{2t})^2}\]
this was my derivative

Answer 17

did you use the product rule on the first two functions?

Answer 18

no

Answer 19

maybe you should :)

Answer 20

haha yeah

Answer 21

what about the third one

Answer 22

3rd is fine

Answer 23

cos(t)-tsin(t)+sin(t)+tcos(t)

Answer 24

what is that

Answer 25

thatas the first 2

Answer 26

remember ..you take the derivative of each of the 3 functions...square each of them...then add...then take square root

Answer 27

i dont know messing up right now

Answer 28

your two derivatives are fine...you just don't add them like you did

Answer 29

I need to go to my office...I'll check back in later on your progress.

Answer 30

remember that those i,j,k mean you have a vector with three components

your derivative is
(cos(t)-tsin(t)) *i + (sin(t)+tcos(t))*j + sqrt(2t)* k

or written as three components of a vector
v= [ (cos(t)-tsin(t)) (sin(t)+tcos(t)) sqrt(2t)]
the magnitude squared is v dot v or v^T v (v transpose times v)