정적분의 뜻과 성질

A ball is thrown upward from a height of 1.5 m at an initial speed of 40 m/sec. Acceleration resulting from gravity is −9.8 m/sec². Neglecting air resistance, solve for the velocity $v(t)$ and the height $h(t)$ of the ball t seconds after it is thrown and before it returns to the ground.

A ball is thrown upward from a height of 3 m at an initial speed of 60 m/sec. Acceleration resulting from gravity is −9.8 m/sec². Neglecting air resistance, solve for the velocity $v(t)$ and the height $h(t)$ of the ball t seconds after it is thrown and before it returns to the ground.

A spherical balloon is being inflated at a constant rate. If the volume of the balloon changes from 36π in.³ to 288π in.³ between time $t=30$ and $t=60$ seconds, find the net change in the radius of the balloon during that time.

Water flows into a conical tank with cross-sectional area πx² at height x and volume $\frac{\pi {x}^{3}}{3}$ up to height x. If water flows into the tank at a rate of 1 m³/min, find the height of water in the tank after 5 min. Find the change in height between 5 min and 10 min.

A horizontal cylindrical tank has cross-sectional area $A(x)=4(6x-{x}^{2}){m}^{2}$ at height x meters above the bottom when $x\le 3.$

a. The volume V between heights a and b is ${\int }_{a}^{b}A(x)dx.$ Find the volume at heights between 2 m and 3 m. b. Suppose that oil is being pumped into the tank at a rate of 50 L/min. Using the chain rule, $\frac{dx}{dt}=\frac{dx}{dV}\,\frac{dV}{dt},$ at how many meters per minute is the height of oil in the tank changing, expressed in terms of x, when the height is at x meters? c. How long does it take to fill the tank to 3 m starting from a fill level of 2 m?

The following table lists the electrical power in gigawatts—the rate at which energy is consumed—used in a certain city for different hours of the day, in a typical 24-hour period, with hour 1 corresponding to midnight to 1 a.m.

Find the total amount of energy in gigawatt-hours (gW-h) consumed by the city in a typical 24-hour period.

The average residential electrical power use (in hundreds of watts) per hour is given in the following table.

a. Compute the average total energy used in a day in kilowatt-hours (kWh). b. If a ton of coal generates 1842 kWh, how long does it take for an average residence to burn a ton of coal? c. Explain why the data might fit a plot of the form $p(t)=11.5-7.5\sin \left(\frac{\pi t}{12}\right).$

The data in the following table are used to estimate the average power output produced by Peter Sagan for each 15-min interval of Stage 1 of the 2012 Tour de France.

Average Power Output

Source: sportsexercisengineering.com

Estimate the net energy used in kilojoules, noting that 1W = 1 J/s.

Newton’s law of gravity states that the gravitational force exerted by an object of mass M and one of mass m with centers that are separated by a distance r is $F=G\frac{mM}{{r}^{2}},$ with G an empirical constant $G=6.67x{10}^{-11}\,{m}^{3}/(kg\cdot {s}^{2}).$ The work done by a variable force over an interval $[a,b]$ is defined as $W={\int }_{a}^{b}F(x)dx.$ If Earth has mass $5.97219\,\times \,{10}^{24}$ and radius 6371 km, compute the amount of work to elevate a polar weather satellite of mass 1400 kg to its orbiting altitude of 850 km above Earth.

John is a 25-year old man who weighs 160 lb. He burns $500-50t$ calories/hr while riding his bike for t hours. If an oatmeal cookie has 55 cal and John eats cookies at a rate of 4t calories during the tth hour, how many net calories has he lost after 3 hours riding his bike?

Sandra is a 25-year old woman who weighs 120 lb. She burns $300-50t$ cal/hr while walking on her treadmill. Her caloric intake from drinking Gatorade is 100t calories/hour during the tth hour. What is her net decrease in calories after walking for 3 hours?

A motor vehicle has a maximum efficiency of 33 mpg at a cruising speed of 40 mph. The efficiency drops at a rate of 0.1 mpg/mph between 40 mph and 50 mph, and at a rate of 0.4 mpg/mph between 50 mph and 80 mph. What is the efficiency in miles per gallon if the car is cruising at 50 mph? What is the efficiency in miles per gallon if the car is cruising at 80 mph? If gasoline costs $3.50/gal, what is the cost of fuel to drive 50 mi at 40 mph, at 50 mph, and at 80 mph?

Although some engines are more efficient at given a horsepower than others, on average, fuel efficiency decreases with horsepower at a rate of $1/25$ mpg/horsepower. If a typical 50-horsepower engine has an average fuel efficiency of 32 mpg, what is the average fuel efficiency of an engine with the following horsepower: 150, 300, 450?

[T] The following table lists the 2013 schedule of federal income tax versus taxable income.

Federal Income Tax Versus Taxable Income

Source: http://www.irs.gov/pub/irs-prior/i1040tt--2013.pdf.

Suppose that Steve just received a $10,000 raise. How much of this raise is left after federal taxes if Steve’s salary before receiving the raise was $40,000? If it was $90,000? If it was $385,000?

[T] The following table provides hypothetical data regarding the level of service for a certain highway.

a. Plot vehicles per hour per lane on the x-axis and highway speed on the y-axis. b. Compute the average decrease in speed (in miles per hour) per unit increase in congestion (vehicles per hour per lane) as the latter increases from 600 to 1000, from 1000 to 1500, and from 1500 to 2100. Does the decrease in miles per hour depend linearly on the increase in vehicles per hour per lane? c. Plot minutes per mile (60 times the reciprocal of miles per hour) as a function of vehicles per hour per lane. Is this function linear?

[T] The graph below plots the quadratic $p(t)=6.48{t}^{2}-80.3\,1t+585.69$ against the data in preceding table, normalized so that $t=0$ corresponds to 1963. Estimate the average number of bald eagles per year present for the 37 years by computing the average value of p over $[0,37].$

[T] The graph below plots the cubic $p(t)=0.07{t}^{3}+2.42{t}^{2}-25.63t+521.23$ against the data in the preceding table, normalized so that $t=0$ corresponds to 1963. Estimate the average number of bald eagles per year present for the 37 years by computing the average value of p over $[0,37].$

[T] Suppose you go on a road trip and record your speed at every half hour, as compiled in the following table. The best quadratic fit to the data is $q(t)=5{x}^{2}-11x+49,$ shown in the accompanying graph. Integrate q to estimate the total distance driven over the 3 hours.

[T] The accompanying graph plots the best quadratic fit, $a(t)=-0.70{t}^{2}+1.44t+10.44,$ to the data from the preceding table. Compute the average value of $a(t)$ to estimate the average acceleration between $t=0$ and $t=5.$

[T] Using your acceleration equation from the previous exercise, find the corresponding velocity equation. Assuming the initial velocity is 65 mph, find the velocity at time $t=0.$