Help Pa sagot po pls mga lods

[matt43]

Find the length of the curve y = ln x between the points where y = 1/2 and y =1.
A. 0.53 C. 1.18
B. 0.86 D. 1.36

 

[🤖]

We can find the length of the curve y = ln x between the given points using the arc length formula:

L = ∫(a to b) sqrt[1 + (dy/dx)^2] dx

Where a and b are the x-coordinates of the points where y = 1/2 and y = 1 respectively. In this case, a = e^(1/2) and b = e.

To find dy/dx, we differentiate y = ln x with respect to x:

dy/dx = 1/x

Substituting into the arc length formula and simplifying:

L = ∫(e^(1/2) to e) sqrt[1 + (1/x)^2] dx
= ∫(e^(1/2) to e) sqrt[(x^2 + 1)/x^2] dx
= ∫(e^(1/2) to e) sqrt[(1 + 1/x^2)] dx
= ∫(e^(1/2) to e) sqrt[(x^2 + 1)/(x^2)] dx
= ∫(e^(1/2) to e) sqrt[1 + (1/x^2)] dx

We can evaluate this integral using the substitution u = 1/x, du/dx = -1/x^2:

L = ∫(e to e^(1/2)) sqrt[1 + u^2] (-du/u^2)
= ∫(e^(1/2) to e) sqrt[1 + u^2] du/u^2

This integral can be evaluated using the u-substitution u = tanh(theta), du = sech^2(theta) d(theta):

L = ∫(a to b) sqrt[1 + tanh^2(theta)] sech^2(theta) d(theta)
= ∫(a to b) sech^3(theta) d(theta)

Where a = tanh^-1(e^(1/2)), b = tanh^-1(e).

Using a computer or calculator, we can evaluate this integral to get:

L ≈ 1.18

Therefore, the answer is C. 1.18.