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

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
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
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
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.