- #1

chwala

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- Homework Statement
- see attached

- Relevant Equations
- Integration -Hyperbolic Functions

Just went through this...steps pretty clear. I refreshed on Riemann integrals { sum of rectangles approximate area under curves}. My question is on the highlighted part in Red. The approximation of area under curve may be smaller or larger than the actual value. Thus the inequality may be ##<## or ##>##. Correct?

In the case of this question they chose ##>##. My point is they could have as well chosen to use ##<## with no implications.

My general comments on the ms approach is as follows,

##\cosh^{-1} r = \ln (r + \sqrt {r^2 -1} )## stems from one understanding the following steps

Let ##y = \cosh^{-1} x##

then

##x = \cosh y = \dfrac {e^y + e^{-y}}{2}##

##x= \dfrac{e^y + e^{-y}}{2}##

##2xe^y = e^{2y}+1##

...

##⇒ e^y = x ± \ln (x + \sqrt {x^2 -1} )##

##y = x ± \ln (x + \sqrt {x^2 -1} )##

They also used integration by parts noting that

##u = \cosh^{-1} x##

using

## \cosh^2 y - \sinh^2y = 1## and letting ## y =\cosh^{-1} x## then ##x = \cosh y##

##\dfrac{dy}{dx} = \dfrac{1}{\sinh y}##

##\sinh y = \sqrt{\cosh^2y -1}##

##\dfrac{dy}{dx}= \dfrac{1}{\sqrt{x^2-1)}}##

and lastly,

##\ln (r + \sqrt {r^2 -1} ) > [x \ln r + \sqrt {r^2-1}]_1^N - [\sqrt {x^2-1}]_1^N##

##\ln (r + \sqrt {r^2 -1} ) > N \ln N +\sqrt {N^2-1} -0-\sqrt {N^2-1}+0##

##\ln (r + \sqrt {r^2 -1} ) > N \ln N +\sqrt {N^2-1}-\sqrt {N^2-1}+0##

In the case of this question they chose ##>##. My point is they could have as well chosen to use ##<## with no implications.

My general comments on the ms approach is as follows,

##\cosh^{-1} r = \ln (r + \sqrt {r^2 -1} )## stems from one understanding the following steps

Let ##y = \cosh^{-1} x##

then

##x = \cosh y = \dfrac {e^y + e^{-y}}{2}##

##x= \dfrac{e^y + e^{-y}}{2}##

##2xe^y = e^{2y}+1##

...

##⇒ e^y = x ± \ln (x + \sqrt {x^2 -1} )##

##y = x ± \ln (x + \sqrt {x^2 -1} )##

They also used integration by parts noting that

##u = \cosh^{-1} x##

using

## \cosh^2 y - \sinh^2y = 1## and letting ## y =\cosh^{-1} x## then ##x = \cosh y##

##\dfrac{dy}{dx} = \dfrac{1}{\sinh y}##

##\sinh y = \sqrt{\cosh^2y -1}##

##\dfrac{dy}{dx}= \dfrac{1}{\sqrt{x^2-1)}}##

and lastly,

##\ln (r + \sqrt {r^2 -1} ) > [x \ln r + \sqrt {r^2-1}]_1^N - [\sqrt {x^2-1}]_1^N##

##\ln (r + \sqrt {r^2 -1} ) > N \ln N +\sqrt {N^2-1} -0-\sqrt {N^2-1}+0##

##\ln (r + \sqrt {r^2 -1} ) > N \ln N +\sqrt {N^2-1}-\sqrt {N^2-1}+0##

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