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Area under curves and Limit of a sequence, need help!

namerequired

New member
Feb 6, 2014
4
Hello, I am looking for an help about this, I have very short time to do many of them and those are an example, could someone show me one solution or explain me how to do it?
Thank you if you can help me, I really appreciate.
Francesco.




 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
Judging by the picture (clickable)

[GRAPH]5wib4rocqz[/GRAPH]

the area is
\[
\int_{0}^2(-(x-2)^3+2)-(x^2-2)\,dx
\]

Limits of the form $\lim_{n\to\infty}f(n)^{g(n)}$ are usually easier to compute when the function is represented as $e^{g(n)\ln(f(n))}$. Which ways of finding limits do you know?
 

namerequired

New member
Feb 6, 2014
4
Judging by the picture (clickable)

[GRAPH]5wib4rocqz[/GRAPH]

the area is
\[
\int_{0}^2(-(x-2)^3+2)-(x^2-2)\,dx
\]

Limits of the form $\lim_{n\to\infty}f(n)^{g(n)}$ are usually easier to compute when the function is represented as $e^{g(n)\ln(f(n))}$. Which ways of finding limits do you know?
By substitution, or expanding, or Hospital rule i suppose.

Thank you
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,492
By substitution, or expanding, or Hospital rule i suppose.
I think, the easiest way is to expand $\ln(1+x)$ as $x+o(x)$, but l'Hospital's rule works too. Recall that to apply the rule you need to represent the function as a ratio of two functions that tend both to zero or both to infinity.