Integration problem using inscribed rectangles

[chwala]

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,


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

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

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

 

[docnet]

The approximation of area under curve may be smaller or larger than the actual value. Thus the inequality may be $<$ or $>$. Correct?

The correct inequality is
$$\sum_{i=2}^N\cosh ^{-1}i>\int^N_1 \cosh ^{-1}x dx$$
because the Riemann sum uses the right endpoints of each subinterval and the function is increasing. The inequality could be wrong depending on if it's a left, middle, or trapezoidal Riemann sum.

 

[docnet]

For a general Riemann sum is evaluated over the interval $[a,b]$, if we evaluate the sum using increasingly narrower rectangles, the limit as the width of the rectangles go to zero is exact integral. Specifically using the right endpoint rule, with partition $a=x_1<\cdots <x_n=b$,
$$\lim_{n\to \infty}\sum^{n}_{i=1}f(x_{i+1})(x_{i+1}-x_i)=\int^b_af(x)dx.$$

 

[WWGD]

The above is the meaning/definition of Riemann integrability. The convergence in the sense that as $Lim ||P||_{\rightarrow 0}$, meaning the maximum width of the rectangles becomes indefinitely small, the value of the Riemann Integral becomes indefinitely -close to a value $R$ ; in a" $P-\epsilon$" sense, of the Riemann integral. This is called Net Convergence.

 

[docnet]

The above is the meaning/definition of Riemann integrability. The convergence in the sense that as $Lim ||P||_{\rightarrow 0}$, meaning the maximum width of the rectangles becomes indefinitely small, the value of the Riemann Integral becomes indefinitely -close to a value $R$ ; in a" $P-\epsilon$" sense, of the Riemann integral. This is called Net Convergence.

Just want to say, I noticed that my comment isn't right. I stated the limit as the number of rectangles go to infinity, but I didn't specify that the width of every rectangle goes to zero, or equivalently that $P=\text{max}_{i\in I}\left(x_{i+1}-x_i\right),$ $I=\{1,...,N\}$ goes to zero. I had a feeling that something was off, but didn't notice it right away.

Does the '$P - \epsilon$' sense of convergence mean that, for every $\epsilon>0$, there exists a $P$ such that the corresponding Riemann sum and the value $R$ differ by $\epsilon$?

 

[WWGD]

Just want to say, I noticed that my comment isn't right. I stated the limit as the number of rectangles go to infinity, but I didn't specify that the width of every rectangle goes to zero, or equivalently that $P=\text{max}_{i\in I}\left(x_{i+1}-x_i\right),$ $I=\{1,...,N\}$ goes to zero. I had a feeling that something was off, but didn't notice it right away.

Does the '$P - \epsilon$' sense of convergence mean that, for every $\epsilon>0$, there exists a $P$ such that the corresponding Riemann sum and the value $R$ differ by $\epsilon$?

Let $R(f)$ be the Riemann sum associated with $f$.Then the Riemann Integral of $f$ converges to $R$ means that for every $\epsilon >0$, there is a$\delta$ , so that $|R(f) -R|<\epsilon$ for any partition $P$ with $||P||<\delta$.

 

[pasmith]

This isn't a question on Riemann sums; it's a question on the integral comparison test for the convergence of a series.

 

[WWGD]

This isn't a question on Riemann sums; it's a question on the integral comparison test for the convergence of a series.

I'm answering a question that was asked.

 

[pasmith]

I'm answering a question that was asked.

I'm commenting on the OP's choice of thread title.

 

[WWGD]

I'm commenting on the OP's choice of thread title.

Ok, my bad.

 

[Mark44]

I'm commenting on the OP's choice of thread title.

I changed the thread title to "Integration problem using inscribed rectangles." I hope this better describes what the thread is about. Possibly @pasmith's recommendation of "integral comparison test" might be a better choice, though.

 

[WWGD]

I changed the thread title to "Integration problem using inscribed rectangles." I hope this better describes what the thread is about. Possibly @pasmith's recommendation of "integral comparison test" might be a better choice, though.

Then we may have to go through several likes of " proof this solution" , " Avance Calculas" I've seen. Should I point them out to staff?

 

[Mark44]

Then we may have to go through several likes of " proof this solution" , " Avance Calculas" I've seen. Should I point them out to staff?

I regularly edit thread titles with gross typos, but haven't seen any lately like the ones you cited. @berkeman also changes thread titles if they don't match the thread content very well or aren't descriptive.
If you see any that you think ought to be changed, of relatively recent threads, yes, go ahead and report them.

 

[docnet]

Let $R(f)$ be the Riemann sum associated with $f$.Then the Riemann Integral of $f$ converges to $R$ means that for every $\epsilon >0$, there is a$\delta$ , so that $|R(f) -R|<\epsilon$ for any partition $P$ with $||P||<\delta$.

If I understand correctly, then $||P||$ means $$\sqrt{\sum_i\left(\Delta x_i^2\right)}?$$ Thanks

 

[pasmith]

If I understand correctly, then $||P||$ means $$\sqrt{\sum_i\left(\Delta x_i^2\right)}?$$ Thanks

$\|P\| = \max_i \Delta_i$.