적분 전체 문제

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?

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?

[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 $t=0.$

[T] Using your velocity equation from the previous exercise, find the corresponding distance equation, assuming your initial distance is 0 mi. How far did you travel while you accelerated your car? (Hint: You will need to convert time units.)

If the average cost per gigabyte of RAM in 2010 is $12, find the average cost per gigabyte of RAM in 1980.

The average cost per gigabyte of RAM can be approximated by the function $C(t)=8,500,000{(0.65)}^{t},$ where $t$ is measured in years since 1980, and $C$ is cost in US$. Find the average cost per gigabyte of RAM for 1980 to 2010.

Find the average cost of 1GB RAM for 2005 to 2010.

The velocity of a bullet from a rifle can be approximated by $v(t)=6400{t}^{2}-6505t+2686,$ where $t$ is seconds after the shot and $v$ is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: $0\le t\le 0.5.$ What is the total distance the bullet travels in 0.5 sec?

What is the average velocity of the bullet for the first half-second?

An object is thrown straight up with a velocity, in ft/s, given by $v(t) = -32t+64$, where $t$ is in seconds, from a height of 48 feet.

(a) What is the object's maximum velocity? (b) What is the object's maximum displacement? (c) When does the maximum displacement occur? (d) When will the object reach a height of 0? (Hint: find when the displacement is $-48$ft.)

An object is thrown straight up with a velocity, in ft/s, given by $v(t) = -32t+96$, where $t$ is in seconds, from a height of 64 feet.

(a) What is the object's initial velocity? (b) When is the object's displacement 0? (c) How long does it take for the object to return to its initial height? (d) When will the object reach a height of 210 feet?

In Exercises 41–46, a velocity function of an object moving along a straight line is given. Find the displacement of the object over the given time interval.

$v(t) = -32t+20$ft/s on $[0,5]$