91Ó°ÊÓ

Area and Volume In Exercises 67 and 68, consider the region satisfying the inequalities. (a) Find the area of the region. (b) Find the volume of the solid generated by revolving the region about the \(x\)-axis. (c) Find the volume of the solid generated by revolving the region about the \(y\)-axis. $$y \leq \frac{1}{x^{2}}, y \geq 0, x \geq 1$$

Propulsion In Exercises 71 and 72, use the weight of the rocket to answer each question.(Use 4000 miles as the radius of Earth and do not consider the effect of air resistance.) (a) How much work is required to propel the rocket an unlimited distance away from Earth's surface? (b) How far has the rocket traveled when half of the total work has occurred? 5-metric-ton rocket

HOW DO YOU SEE IT? The graph shows the probability density function for a car brand that has a mean fuel efficiency of 26 miles per gallon and a standard deviation of 2.4 miles per gallon. (a) Which is greater, the probability of choosing a car at random that gets between 26 and 28 miles per gallon or the probability of choosing a car at random that gets between 22 and 24 miles per gallon? (b) Which is greater, the probability of choosing a car at random that gets between 20 and 22 miles per gallon or the probability of choosing a car at random that gets at least 30 miles per gallon? $$ \begin{array}{l}{\text { Capitalized cost In Exercises } 77 \text { and } 78 \text { , find the capitalized }} \\ {\text { cost } C \text { of an asset (a) for } n=5 \text { years, (b) for } n=10 \text { years, and }} \\ {\text { (c) forever. The capitalized cost is given by }} \\ {C=C_{0}+\int_{0}^{n} c(t) e^{-r t} d t}\end{array} $$ $$\begin{array}{l}{\text { where } C_{0} \text { is the original investment, } t \text { is the time in years, } r \text { is }} \\ {\text { the annual interest rate compounded continuously, and } c(t) \text { is }} \\ {\text { the annual cost of maintenance. }}\end{array}$$ $$\begin{array}{l}{C_{0}=\$ 700,000} \\ {c(t)=\$ 25,000(1+0.08 t)} \\\ {r=0.06}\end{array}$$

In Exercises 91-98, find the Laplace Transform of the function. $$f(t)=t$$