I have a region $R$ defined by $y=x^2$, $y=2+x$ and $x=0$.
How would be the integrals (no need to develop them) of the solid obtained by the revolution of $R$:
$a)$ around $x$-axis integrating in relation to $1)$ $x$ and $2)$ $y$?
$b)$ around $y$-axis integrating in relation to $1)$ $x$ and $2)$ $y$?
I just started studying calculus by myself and I would like to check some answers.
In the item a.1 I found $\int_0^2\pi|x^4-(1+x)^2| dx$. Is that correct? How would be the others?
No, that is not correct.
The correct formula is $V = \pi \int_a^b [f(x)]^2 \ dx$. The limits of integration are $x = 0, 2$ as you have said, but $f(x) = \text{top - bottom} = (2+x) - x^2$.