Bajo la superficie \( z=1+x^{2} y^{2} \) y arriba de la región acotada \( x=y^{2} \) y \( x=4 \)

2) La ecuación del movimiento de una particula está dada por: \( s(t)=t^{3}-3 t \) Encontrar: a) La velocidad como función de \( t \) b) La aceleración después de 4 segundos. c) la aceleración cuando la velocidad es cero.

Use a Riemann sum to approximate the area under the graph of \( f(x)=4 e^{-x} \) on the interval \( 0 \leq x \leq 1 \) using \( n=5 \) subintervals with the selected points as the right end points. The area is approximately \( \square \) units. (Round to six decimal places as needed.)

b Hence show that \( \int_{2}^{3} \ln y \mathrm{~d} y=\ln \left(\frac{27}{4 \mathrm{e}}\right) \)

Find the average rate of change of \( h(x)=-2 x^{2}-5 x \) from \( x=-4 \) to \( x=1 \). Simplify your answer as much as possible.

A national fast food company has just analyzed the effect of the latest sales promotion. It has determined that the sales decay for its milkshakes is given by \( S=285,000 e^{-0.7 x} \), where \( S \) is the weekly sales and \( x \) is the number of weeks that have passes since the end of the sales promotion. a) Determine the sales 2 weeks after the end of the promotion: Round answer to 2 decimal places. Sales \( =\$ \) b) If no new campaign is started, how many weeks after the end of the promotion will sales drop below 30,000 ? Round answer to 1 decimal place. Num. weeks =