Question

Find the area bounded by the curve x=t-1/t, y=t=1/t, y=65/8

Answer 1

Is that \[x= t - \frac{ 1 }{ t }~~~y=t-\frac{ 1 }{ t }~~~y=\frac{ 65 }{ 8 }\]

Answer 2

@Astrophysics Yes it looks like it

Answer 3

y=t=

Answer 4

y could be t+1/t

since, + and = are on the same key of the keyboard

Answer 5

I think you're right hartnn! So areas in parametric curves we have \[\int\limits_{a}^{b}g(t)f'(t) dt\]

Answer 6

now to see where the curve touches the line we set \[t+\frac{ 1 }{ t } = \frac{ 65 }{ 8 }\] solving for t will tell us at which time our curve will touch the line, then you can make a graph to make your curve.

So we get our time to be \[t=\frac{ 1 }{ 8 } ~~~and ~~~ t = 8\]

Answer 7

To find our values for x, you can make a graph as I suggested (just too keep your values and see what's going on), or you can just do when t = 1/8 \[x= t-\frac{ 1 }{ t } \implies \frac{ 1 }{ 8 } - \frac{ 1 }{ \frac{ 1 }{ 8 } } = ...\]
\[x= 8-\frac{ 1 }{ 8 } = ...\]

You'll then have your x,y,t values and you should be able to graph it, that's the major part, then setting up the integral shouldn't be too bad.

Answer 8

I don't plan on doing this whole thing for you, so if you're there, please ask for help.

Answer 9

Alternatively, if you don't like parametric curves, you may try eliminating the parameter easily by noticing that
\[(a+b)^2-(a-b)^2 = 4ab\]

Answer 10

\[y^2 - x^2 = \left(t+\frac{1}{t}\right)^2 - \left(t-\frac{1}{t}\right)^2=4*t*\frac{1}{t} = 4 \]

Answer 11

so the problem is equivalent to finding the area between curves : \(y^2-x^2=4\) and \(y=65/8\)

Answer 12

So much easier xD

Answer 13

Not really, looks both require almost same effort haha!

Answer 14

Ah, I see, I just think it's easier to see the curve with your method...but yeah I think it's equal effort hehe.

Answer 15

still not sure how to get the right answer.

Answer 16

Ok well, doing it the method I suggested, what values did you get for x?

Answer 17

\[x= t-\frac{ 1 }{ t } \implies \frac{ 1 }{ 8 } - \frac{ 1 }{ \frac{ 1 }{ 8 } } = ...\] and \[x= 8-\frac{ 1 }{ 8 } = ...\]

Answer 18

We'll go over this step by step, so just find what x's are and we can continue :)