MHB Area under curves and Limit of a sequence,

[namerequired]

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]

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]

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]

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.