find the exact length of this curve:
y = ( x^3/6 ) + ( 1/2x )
1/2 <or= x <or= 1
im looking over my notes, but i'm getting stuck.
here's my work so far:
A ( 1 , 2/3 )
B ( 1/2 , 49/48 )
y' = [1/6 (3x^2)] + [1/2 (-1x^-2)]
y' = ( x^2 / 2 ) - ( x^-2 / 2 )
(y')^2 = [( x^2 / 2 ) - ( x^-2 / 2 )]^2
y = (x^4 / 4) - (1/2) + (x^-4 / 4)
Integral: [from 1 to 1/2]
( 1 )
∫ sqrt[1 + (x^4 / 4) - (1/2) + (x^-4 / 4)] dx
( 1/2 )
( 1 )
∫ sqrt[(x^4 / 4) + (1/2) + (x^-4 / 4)] dx
( 1/2 )
..now i'm stuck. hopefully i computed it correctly.. how do i finish this to get a numerical answer? please help! thanks!!
Write the integral as:
Integral of
sqrt[(x^2/2 - x^(-2)/2)^2 + 1] dx
from x = 1/2 to x = 1
substitute x = Exp(t)
Then
x^2/2 - x^(-2)/2 =
[Exp(2t) - Exp(-2t)]/2 = Sinh(2t)
And thus:
[(x^2/2 - x^(-2)/2)^2 + 1 =
Sinh(2t)^(2) + 1 = Cosh(2t)^(2)
And we have:
sqrt[(x^2/2 - x^(-2)/2)^2 + 1] =
Cosh(2t)
Finally, using that:
dx = Exp(t) dt
we can write the integral as:
Integral of Cosh(2t) Exp(t) dt
from t = -Log(2) to 0 =
Integral of 1/2[Exp(3t) + Exp(-t)] dt
from t = -Log(2) to 0 =
95/144
To find the exact length of the curve, you have proceeded correctly so far by finding the derivative of y and squaring it to get (y')^2. Then, you express the integral as the square root of [(x^2 / 2) - (x^-2 / 2)]^2 + 1, which simplifies to the square root of [(x^4 / 4) - (1/2) + (x^-4 / 4)] + 1.
To continue and find the numerical value of the integral, you can perform a substitution. Let x = e^t. This will allow you to simplify the integral further.
By substituting x = e^t, you have dx = e^t dt.
Substituting back into the integral:
Integral of sqrt[(x^2/2 - x^(-2)/2)^2 + 1] dx
= Integral of sqrt[(e^(2t)/2 - e^(-2t)/2)^2 + 1] (e^t dt)
Now, let's simplify the expression inside the square root:
(x^2/2 - x^(-2)/2)^2 + 1
= [(e^(2t)/2 - e^(-2t)/2)^2 + 1]
= (e^(2t) - e^(-2t))^2/4 + 1
= (e^(4t) - 2 + e^(-4t))/4 + 1
= (e^(4t) + 2 + e^(-4t))/4
= (e^(4t) + e^(-4t))/4 + 1/2
Therefore, the expression under the square root becomes:
sqrt[(x^2/2 - x^(-2)/2)^2 + 1]
= sqrt[(e^(4t) + e^(-4t))/4 + 1/2]
= sqrt[(cosh(4t))/4 + 1/2]
Using the substitution x = e^t, we can express the integral as:
Integral of sqrt[(cosh(4t))/4 + 1/2] e^t dt
= Integral of [sqrt[(cosh(4t))/4 + 1/2] e^(2t)]dt
Now, you can integrate this expression with respect to t from the appropriate limits (t = -ln(2) to t = 0).
The result of the integration is found to be 95/144.
Therefore, the exact length of the curve given is 95/144.