적분 전체 문제

An icosahedron is a Platonic solid with a surface that consists of 20 equilateral triangles. By how much does the surface area of an icosahedron increase as the side length of each triangle doubles from a unit to 2a units?

Write an integral that quantifies the increase in the volume of a cube when the side length doubles from s unit to 2s units and evaluate the integral.

Write an integral that quantifies the increase in the surface area of a sphere as its radius doubles from R unit to 2R units and evaluate the integral.

Write an integral that quantifies the increase in the volume of a sphere as its radius doubles from R unit to 2R units and evaluate the integral.

Suppose that a particle moves along a straight line with velocity $v(t)=4-2t,$ where $0\le t\le 2$ (in meters per second). Find the displacement at time t and the total distance traveled up to $t=2.$

Suppose that a particle moves along a straight line with velocity defined by $v(t)={t}^{2}-3t-18,$ where $0\le t\le 6$ (in meters per second). Find the displacement at time t and the total distance traveled up to $t=6.$

Suppose that a particle moves along a straight line with velocity defined by $v(t)=|2t-6|,$ where $0\le t\le 6$ (in meters per second). Find the displacement at time t and the total distance traveled up to $t=6.$

Suppose that a particle moves along a straight line with acceleration defined by $a(t)=t-3,$ where $0\le t\le 6$ (in meters per second). Find the velocity and displacement at time t and the total distance traveled up to $t=6$ if $v(0)=3$ and $d(0)=0.$

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?