Oil leaks out of a tanker at a rate of r = f(t) gallons perminute, where t is in

If f(t) is measured in dollars per year and t is measured in years, what are the units of - b a f(t) dt?

If f(t) is measured in meters/second2 and t is measured in seconds, what are the units of - b a f(t) dt?

If f(x) is measured in pounds and x is measured in feet, what are the units of - b a f(x) dx?

In Exercises 47, explain in words what the integral represents and give units. - 3 1 v(t) dt, where v(t) is velocity in meters/sec and t is time in seconds.

In Exercises 47, explain in words what the integral represents and give units. 6 0 a(t) dt, where a(t) is acceleration in km/hr2 and t is time in hours.

In Exercises 47, explain in words what the integral represents and give units. - 2011 2005 f(t) dt, where f(t) is the rate at which the worlds population is growing in year t, in billion people per year.

In Exercises 47, explain in words what the integral represents and give units. - 5 0 s(x) dx, where s(x) is rate of change of salinity (salt concentration) in gm/liter per cm in sea water, and where x is depth below the surface of the water in cm

For the two cars in Example 3, page 291, estimate: (a) The distances moved by car 1 and car 2 during the first minute. (b) The time at which the two cars have gone the same distance.

In Exercises 914, let f(t) = F (t). Write the integral - b a f(t) dt and evaluate it using the Fundamental Theorem of Calculus. F(t) = t 2; a = 1, b = 3

In Exercises 914, let f(t) = F (t). Write the integral - b a f(t) dt and evaluate it using the Fundamental Theorem of Calculus. F(t)=3t 2 + 4t; a = 2, b = 5

In Exercises 914, let f(t) = F (t). Write the integral - b a f(t) dt and evaluate it using the Fundamental Theorem of Calculus. F(t) = ln t; a = 1, b = 5

In Exercises 914, let f(t) = F (t). Write the integral - b a f(t) dt and evaluate it using the Fundamental Theorem of Calculus. F(t) = sin t; a = 0, b = /2

In Exercises 914, let f(t) = F (t). Write the integral - b a f(t) dt and evaluate it using the Fundamental Theorem of Calculus. F(t)=7 4t ; a = 2, b = 3

In Exercises 914, let f(t) = F (t). Write the integral - b a f(t) dt and evaluate it using the Fundamental Theorem of Calculus. F(t) = tan t; a = 0, b =

a) Differentiate x3 + x. (b) Use the Fundamental Theorem of Calculus to find , 2 0 (3x2 + 1) dx.

(a) What is the derivative of sin t? (b) The velocity of a particle at time t is v(t) = cos t. Use the Fundamental Theorem of Calculus to find the total distance traveled by the particle between t = 0 and t = /2.

(a) If F(t) = 1 2 sin2 t, find F (t). (b) Find , 0.4 0.2 sin t cos t dt two ways: (i) Numerically. (ii) Using the Fundamental Theorem of Calculus.

(a) If F(x) = ex2 , find F (x). (b) Find , 1 0 2xex2 dx two ways: (i) Numerically. (ii) Using the Fundamental Theorem of Calculus.

Pollution is removed from a lake on day t at a rate of f(t) kg/day. (a) Explain the meaning of the statement f(12) = 500. (b) If - 15 5 f(t) dt = 4000, give the units of the 5, the 15, and the 4000. (c) Give the meaning of - 15 5 f(t) dt = 4000.

Oil leaks out of a tanker at a rate of r = f(t) gallons per minute, where t is in minutes. Write a definite integral expressing the total quantity of oil which leaks out of the tanker in the first hour.

Water is leaking out of a tank at a rate of R(t) gallons/hour, where t is measured in hours. (a) Write a definite integral that expresses the total amount of water that leaks out in the first two hours. (b) In Figure 5.42, shade the region whose area represents the total amount of water that leaks out in the first two hours. (c) Give an upper and lower estimate of the total amount of water that leaks out in the first two hours.

As coal deposits are depleted, it becomes necessary to strip-mine larger areas for each ton of coal. Figure 5.43 shows the number of acres of land per million tons of coal that will be defaced during strip-mining as a function of the number of million tons removed, starting from the present day. (a) Estimate the total number of acres defaced in extracting the next 4 million tons of coal (measured from the present day). Draw four rectangles under the curve, and compute their area. (b) Re-estimate the number of acres defaced using rectangles above the curve. (c) Use your answers to parts (a) and (b) to get a better estimate of the actual number of acres defaced.

The rate at which the worlds oil is consumed (in billions of barrels per year) is given by r = f(t), where t is in years and t = 0 is the start of 2004. (a) Write a definite integral representing the total quantity of oil consumed between the start of 2004 and the start of 2009. (b) Between 2004 and 2009, the rate was modeled by r = 32e0.05t . Using a left-hand sum with five subdivisions, find an approximate value for the total quantity of oil consumed between the start of 2004 and the start of 2009. (c) Interpret each of the five terms in the sum from part (b) in terms of oil consumption.

A bungee jumper leaps off the starting platform at time t = 0 and rebounds once during the first 5 seconds. With velocity measured downward, for t in seconds and 0 t 5, the jumpers velocity is approximated5 by v(t) = 4t 2 + 16t meters/sec. (a) How many meters does the jumper travel during the first five seconds? (b) Where is the jumper relative to the starting position at the end of the five seconds? (c) What does - 5 0 v(t) dt represent in terms of the jump?

An old rowboat has sprung a leak. Water is flowing into the boat at a rate, r(t), given in the table. (a) Compute upper and lower estimates for the volume of water that has flowed into the boat during the 15 minutes. (b) Draw a graph to illustrate the lower estimate.

Annual coal production in the US (in billion tons per year) is given in the table.6 Estimate the total amount of coal produced in the US between 1997 and 2009. If r = f(t) is the rate of coal production t years since 1997, write an integral to represent the 19972009 coal production.

The amount of waste a company produces, W, in tons per week, is approximated by W = 3.75e0.008t , where t is in weeks since January 1, 2005. Waste removal for the company costs $15/ton. How much does the company pay for waste removal during the year 2005?

A two-day environmental cleanup started at 9 am on the first day. The number of workers fluctuated as shown in Figure 5.44. If the workers were paid $10 per hour, how much was the total personnel cost of the cleanup?

Suppose in Problem 28 that the workers were paid $10 per hour for work during the time period 9 am to 5 pm and were paid $15 per hour for work during the rest of the day. What would the total personnel costs of the cleanup have been under these conditions?

A warehouse charges its customers $5 per day for every 10 cubic feet of space used for storage. Figure 5.45 records the storage used by one company over a month. How much will the company have to pay?

A cup of coffee at 90C is put into a 20C room when t = 0. The coffees temperature is changing at a rate of r(t) = 7e0.1t C per minute, with t in minutes. Estimate the coffees temperature when t = 10

Water is pumped out of a holding tank at a rate of 5 5e0.12t liters/minute, where t is in minutes since the pump is started. If the holding tank contains 1000 liters of water when the pump is started, how much water does it hold one hour later? Problems 3334 concern the graph of f in Figure 5.46.

Which is greater, f(0) or f(1)?

List the following in increasing order: f(4) f(2) 2 , f(3) f(2), f(4) f(3).

A force F parallel to the x-axis is given by the graph in Figure 5.47. Estimate the work, W, done by the force, where W = - 16 0 F(x) dx.

Let f(1) = 7, f (t) = e t2 . Use left- and right-hand sums of 5 rectangles each to estimate f(2).

The graph of a continuous function f is given in Figure 5.48. Rank the following integrals in ascending numerical order. Explain your reasons. (i) - 2 0 f(x) dx (ii) - 1 0 f(x) dx (iii) - 2 0 (f(x))1/2 dx (iv) - 2 0 (f(x))2 dx.

The graphs in Figure 5.49 represent the velocity, v, of a particle moving along the x-axis for time 0 t 5. The vertical scales of all graphs are the same. Identify the graph showing which particle: (a) Has a constant acceleration. (b) Ends up farthest to the left of where it started. (c) Ends up the farthest from its starting point. (d) Experiences the greatest initial acceleration. (e) Has the greatest average velocity. (f) Has the greatest average acceleration.

A car speeds up at a constant rate from 10 to 70 mph over a period of half an hour. Its fuel efficiency (in miles per gallon) increases with speed; values are in the table. Make lower and upper estimates of the quantity of fuel used during the half hour.

In Problems 4041, oil is pumped from a well at a rate of r(t) barrels per day. Assume that t is in days, r (t) < 0 and t0 > 0. What does the value of - t0 0 r(t) dt tells us about the oil well?

In Problems 4041, oil is pumped from a well at a rate of r(t) barrels per day. Assume that t is in days, r (t) < 0 and t0 > 0. Rank in order from least to greatest: , 2t0 0 r(t) dt, , 2t0 t0 r(t) dt, , 3t0 2t0 r(t) dt.

Height velocity graphs are used by endocrinologists to follow the progress of children with growth deficiencies. Figure 5.50 shows the height velocity curves of an average boy and an average girl between ages 3 and 18. (a) Which curve is for girls and which is for boys? Explain how you can tell. (b) About how much does the average boy grow between ages 3 and 10? (c) The growth spurt associated with adolescence and the onset of puberty occurs between ages 12 and 15 for the average boy and between ages 10 and 12.5 for the average girl. Estimate the height gained by each average child during this growth spurt. (d) When fully grown, about how much taller is the average man than the average woman? (The average boy and girl are about the same height at age 3.)

In Problems 4345, evaluate the expressions using Table 5.8. Give exact values if possible; otherwise, make the best possible estimates using left-hand Riemann sums. , 0.5 0 f(t) dt

In Problems 4345, evaluate the expressions using Table 5.8. Give exact values if possible; otherwise, make the best possible estimates using left-hand Riemann sums. , 0.5 0.2 g (t) dt

In Problems 4345, evaluate the expressions using Table 5.8. Give exact values if possible; otherwise, make the best possible estimates using left-hand Riemann sums. , 0.3 0 g (f(t)) dt

In Problems 4648, let C(n) be a citys cost, in millions of dollars, for plowing the roads, when n inches of snow have fallen. Let c(n) = C (n). Evaluate the expressions and interpret your answers in terms of the cost of plowing snow, given c  (n) < 0, , 15 0 c(n) dn = 7.5, c(15) = 0.7, c(24) = 0.4, C(15) = 8, C(24) = 13. , 24 15 c(n) dn

In Problems 4648, let C(n) be a citys cost, in millions of dollars, for plowing the roads, when n inches of snow have fallen. Let c(n) = C (n). Evaluate the expressions and interpret your answers in terms of the cost of plowing snow, given c  (n) < 0, , 15 0 c(n) dn = 7.5, c(15) = 0.7, c(24) = 0.4, C(15) = 8, C(24) = 13. C(0)

In Problems 4648, let C(n) be a citys cost, in millions of dollars, for plowing the roads, when n inches of snow have fallen. Let c(n) = C (n). Evaluate the expressions and interpret your answers in terms of the cost of plowing snow, given c  (n) < 0, , 15 0 c(n) dn = 7.5, c(15) = 0.7, c(24) = 0.4, C(15) = 8, C(24) = 13. c(15) + , 24 15 c  (n) dn

Problems 4951 refer to a May 2, 2010, article:7 The crisis began around 10 am yesterday when a 10-foot wide pipe in Weston sprang a leak, which worsened throughout the afternoon and eventually cut off Greater Boston from the Quabbin Reservoir, where most of its water supply is stored. . . Before water was shut off to the ruptured pipe [at 6:40 pm], brown water had been roaring from a massive crater [at a rate of] 8 million gallons an hour rushing into the nearby Charles River. Let r(t) be the rate in gallons/hr that water flowed from the pipe t hours after it sprang its leak. Which is larger: , 2 0 r(t) dt or , 4 2 r(t) dt?

Problems 4951 refer to a May 2, 2010, article:7 The crisis began around 10 am yesterday when a 10-foot wide pipe in Weston sprang a leak, which worsened throughout the afternoon and eventually cut off Greater Boston from the Quabbin Reservoir, where most of its water supply is stored. . . Before water was shut off to the ruptured pipe [at 6:40 pm], brown water had been roaring from a massive crater [at a rate of] 8 million gallons an hour rushing into the nearby Charles River. Let r(t) be the rate in gallons/hr that water flowed from the pipe t hours after it sprang its leak. Which is larger: , 4 0 r(t) dt or 4r(4)?

Problems 4951 refer to a May 2, 2010, article:7 The crisis began around 10 am yesterday when a 10-foot wide pipe in Weston sprang a leak, which worsened throughout the afternoon and eventually cut off Greater Boston from the Quabbin Reservoir, where most of its water supply is stored. . . Before water was shut off to the ruptured pipe [at 6:40 pm], brown water had been roaring from a massive crater [at a rate of] 8 million gallons an hour rushing into the nearby Charles River. Let r(t) be the rate in gallons/hr that water flowed from the pipe t hours after it sprang its leak. Give a reasonable overestimate of , 8 0 r(t) dt.

In Problems 5253, explain what is wrong with the statement. If f(t) represents the rate, in lbs per year, at which a dog gains weight t years after it is born, then - 4 0 f(t)dt represents the weight of the dog when the dog is four years old.

In Problems 5253, explain what is wrong with the statement. If f(x) = x the Fundamental Theorem of Calculus states that - 9 4 x dx = 9 4.

In Problems 5455, give an example of: A function f(x) and limits of integration a and b such that - b a f(x) dx = e4 e2

In Problems 5455, give an example of: The graph of a velocity function of a car that travels 200 miles in 4 hours.

True or False? The units for an integral of a function f(x) are the same as the units for f(x).