Question

Find the length of the curve defined by y = 6x^(3/2) + 7 from x = 4 to x = 7.

Answer 1

haha, Same goes with that question.

Answer 2

help bro

Answer 3

first derivative the function with respect to y.
\[\frac{ 18 }{ 2 }\times x ^{\frac{ 1 }{ 2 }}\]

Answer 4

simplify, 9 sqrt(x)

Answer 5

why is it x^1/2

Answer 6

oh cause -1?

Answer 7

to derivative, exponential functions, bring down the original exponent, then minus one the exponent to find the new exponent

Answer 8

\[\int_4^7\sqrt{1+(f'(x))^2}dx\]

Answer 9

i need the steps though

Answer 10

the problem worked out

Answer 11

\[f(x)=6x^{\frac{3}{2}}\]
\[f'(x)=9x^{\frac{1}{2}}\]
\[(f'(x))^2=81x\]
\[\int_4^7\sqrt{1+81x}dx\]

Answer 12

now what

Answer 13

integrate?

Answer 14

yes.

Answer 15

i need help though

Answer 16

try \(u=1+81x\)

Answer 17

whats that suppse to do

Answer 18

\[\int\limits_{4}^{7}\sqrt{1+81x}dx\]

Answer 19

but what do i do from there

Answer 20

let 1+81x = u then du = 81dx

Answer 21

substitute

Answer 22

ok so...

Answer 23

\[\frac{ 1 }{ 81 } \int\limits_{4}^{7}u ^{\frac{ 1 }{ 2 }}du\]

Answer 24

why did you pull out 1/81 and was it u^3/2+1

Answer 25

? help

Answer 26

look again from the original equation, you should substitute dx with du/81 right?

Answer 27

can u draw it

Answer 28

ist't it that you have the integral of the sqrt of (1 + 81x) dx? look above

Answer 29

yes the part that satelitte drew

Answer 30

thats where im at, can u work with me from there

Answer 31

ok, then you let 1+81x = u
got it?

Answer 32

then du is going to be equal to 81dx
got it?

Answer 33

yes

Answer 34

ok from there, dx = du/81
right?

Answer 35

yes

Answer 36

then substitute you will arrive with my final equation a while ago
got it?

Answer 37

can u show me

Answer 38

?

Answer 39

replace 1+81x with u
then dx with du/81

Answer 40

so its sqrt(1+81x) * 81 dx?

Answer 41

its going to be sqrt(u) du / 81

Answer 42

ok now what

Answer 43

then you can now integrate.

Answer 44

i thought we just did , can u just stay on my problem and help me

Answer 45

so you're going to have this final equation after substitution
\[\frac{ 1 }{ 81 } \int\limits_{4}^{7}\sqrt{u}\]

Answer 46

du

Answer 47

now what

Answer 48

integrate
you will arrive at
\[u ^{\frac{ 1 }{ 2 }+1} \div (\frac{ 1 }{2 }+1) \]
evaluated from 4 to 7

Answer 49

where did the 1/81 go?

Answer 50

nevermind so now what

Answer 51

i just forgot to type it, but it will stay there

Answer 52

ok i got it 63?

Answer 53

\[\frac{ 1 }{ 81 } u ^{\frac{ 3 }{ 2 }}\times \frac{ 1 }{ \frac{ 3 }{ 2 } }\]

Answer 54

ya

Answer 55

there you go.

Answer 56

63