Let \( R \) be the region enclosed by the graphs of \( f(x)=4 \cos (x)-3 \) and \( g(x)=4 \sin (x)-4 \). Find the length of the boundary of region \( R \). Use a calculator and round your answer to 3 decimal places.

Determine the general solution of the differential equation \( \frac{dy}{dx} = ky^2 \), where k is a constant.

Evaluar la integral iterada siguiente: \( \int_{0}^{2} \int_{1}^{4}\left(-5 \cdot y^{2}+x^{2}+5\right) d x d y \) Primero, resolver la integral: \( \int_{1}^{4}\left(-5 \cdot y^{2}+x^{2}+5\right) d x=\square \) Luego, integrando con respecto a \( y \). \( \square \)

\( \int_{1}^{2}\left(-2 y^{2}+x^{2}+4\right) d x=\square \) Luego, integrando con respecto a \( y \)

La segunda parcial de la siguiente función \( f(x, y)=3 x-x^{2} y^{2}+2 x^{3} y \) con respecto a \( x \) es: \( 3-2 x \)

solve \( \lim _{x \rightarrow 1} \frac{x-1}{\sqrt{x}-1} \)