[2] Multivariable integral part
(1) Evaluate the following integral by first reversing the order of integration:
$\int_0^1 \int_{\sqrt{x}}^1 \sqrt{y^3 + 1} dydx$
(2) [Derivation process is necessary.] Find the Jacobian $J(r, \theta)$ for the transformation $x = r \cos \theta$, $y = r \sin \theta$.
(3) Find
$\iint_R dxdy$,
where R is the area surrounded by curves $xy = 1$, $xy = 2$, $xy^2 = 1$, and $xy^2 = 3$.
