A mouse moves back and forth in a straight tunnel, attracted to bits of cheddar cheese

The velocity v(t) in Table 5.8 is decreasing, 2 t 12. Using n = 5 subdivisions to approximate the total distance traveled, find (a) Anupperestimate (b) Alowerestimate Table 5.8 t 2 4 6 8 10 12 v(t) 44 42 41 40 37 35

The velocity v(t) in Table 5.9 is increasing, 0 t 12. (a) Find an upper estimate for the total distance traveled using (i) n = 4 (ii) n = 2 (b) Which of the two answers in part (a) is more accurate? Why? (c) Find a lower estimate of the total distance traveled using n = 4. Table 5.9 t 0 3 6 9 12 v(t) 34 37 38 40 45

Use the following table to estimate =25 0 f(x)dx.

If f(t) is measured in miles per hour and t is measured in hours, what are the units of =b a f(t) dt?

Figure 5.80 shows the velocity of a runner for 0 t 8 and the rectangles used to estimate the distance traveled. (a) Do the rectangles represent a left or a right sum? (b) Do the rectangles lead to an upper or a lower estimate? (c) What is the value of n? (d) What is the value of t? (e) Give an approximate value for the estimate. 0 8 18 t Figure 5.80

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

As coal deposits are depleted, it becomes necessary to strip-mine larger areas for each ton of coal. Figure 5.81 shows the number of acres of land permillion 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. 1 2 3 4 5 1 2 3 4 0.2 million tons of coal extracted (measured from present day) acres defaced per million tons Figure 5.81

Find the area under y = x3 + 2 between x = 0 and x = 2. Sketch this area.

Find the area under the graph of y = 10x(3x) between x = 0 and x = 3. Sketch this area.

For Problems 1219, use a calculator or computer to evaluate the integral. < 2 1 2x dx

For Problems 1219, use a calculator or computer to evaluate the integral. < 4 1 (x2 + x) dx

For Problems 1219, use a calculator or computer to evaluate the integral. < 5 1 (3x + 1)2 dx

For Problems 1219, use a calculator or computer to evaluate the integral. < 1 0 41 + t2 dt

For Problems 1219, use a calculator or computer to evaluate the integral. < 2 1 (1.03)t dt

For Problems 1219, use a calculator or computer to evaluate the integral. < 3 2 1 (r + 1)2 dr

For Problems 1219, use a calculator or computer to evaluate the integral. < 3 1 z2 + 1 z dz

For Problems 1219, use a calculator or computer to evaluate the integral. < 3 3 et2 dt

In Problems 2023, find the given area. Between y = x2 and y = x3 for 0 x 1.

In Problems 2023, find the given area. Between y = x1/2 and y = x1/3 for 0 x 1.

In Problems 2023, find the given area. Between y = 3x and y = x2.

In Problems 2023, find the given area. Between y = x and y = x.

Coal gas is produced at a gasworks. Pollutants in the gas are removed by scrubbers, which become less and less efficient as time goes on. The following measurements, made at the start of each month, show the rate at which pollutants are escaping (in tons/month) in the gas: Time (months) 0 1 2 3 4 5 6 Rate pollutants escape 5 7 8 10 13 16 20 (a) Make an overestimate and an underestimate of the total quantity of pollutants that escape during the first month. (b) Make an overestimate and an underestimate of the total quantity of pollutants that escape during the six months.

A student is speeding down Route 11 in his fancy red Porsche when his radar system warns him of an obstacle 400 feet ahead. He immediately applies the brakes, starts to slow down, and spots a skunk in the road directly ahead of him. The black box in the Porsche records the cars speed every two seconds, producing the following table. The speed decreases throughout the 10 seconds it takes to stop, although not necessarily at a constant rate. (a) What is your best estimate of the total distance the students car traveled before coming to rest? (b) Which one of the following statements can you justify from the information given? (i) The car stopped before getting to the skunk. (ii) The black box data is inconclusive. The skunk may or may not have been hit. (iii) The skunk was hit by the car.

The velocity of a particle moving along the x-axis is given by f(t) = 6 2t cm/sec. Use a graph of f(t) to find the exact change in position of the particle from time t = 0 to t = 4 seconds.

A baseball thrown directly upward at 96 ft/sec has velocity v(t) = 96 32t ft/sec at time t seconds. (a) Graph the velocity from t = 0 to t = 6. (b) When does the baseball reach the peak of its flight? How high does it go? (c) How high is the baseball at time t = 5?

A news broadcast in early 1993 said the typical Americans annual income is changing at a rate of r(t) = 40(1.002)t dollars per month, where t is in months from January 1, 1993. How much did the typical Americans income change during 1993?

Two species of plants have the same populations at time t = 0 and the growth rates shown in Figure 5.82. (a) Which species has a larger population at the end of 5 years? At the end of 10 years? (b) Which species do you think has the larger population after 20 years? Explain. 5 10 Species B Species A t (years) new plants per year Figure 5.82

Figure 5.83 represents your velocity, v, on a bicycle trip along a straight road which starts 10 miles from home. Write a paragraph describing your trip: Do you start out going toward or away from home? How long do you continue in that direction and how far are you from home when you turn around? How many times do you change direction? Do you ever get home? Where are you at the end of the four-hour bike ride? 1 2 3 4 20 10 10 20 30 t (hours) v (mph) Away from home Toward home Figure 5.83

Figure 5.84 shows the weight growth rate of a human fetus. (a) What property of a graph of weight as a function of age corresponds to the fact that the function in Figure 5.84 is increasing? (b) Estimate the weight of a baby born in week 40. 8 16 24 32 40 0.1 0.2 age of fetus (wks after last menstruation) weight growth rate (kg/week) Figure 5.84: Rate of increase of fetal weight

A bicyclist is pedaling along a straight road for one hour with a velocity v shown in Figure 5.85. She starts out five kilometers from the lake and positive velocities take her toward the lake. [Note: The vertical lines on the graph are at 10 minute (1/6 hour) intervals.] (a) Does the cyclist ever turn around? If so, at what time(s)? (b) When is she going the fastest? How fast is she going then? Toward the lake or away? (c) When is she closest to the lake? Approximately how close to the lake does she get? (d) When is she farthest from the lake? Approximately how far from the lake is she then?

3 3 f(x) x

3 3 f(x) x

3 3 f(x) x

3 3 f(x) x

Given =0 2 f(x)dx = 4 and Figure 5.86, estimate: (a) =2 0 f(x)dx (b) =2 2 f(x)dx (c) Thetotalshadedarea. 2 2 2 2 f(x) x Figure 5.86

Use a graph of y = 2x2 to explain why =1 1 2x2 dx must be between 0 and 2.

Without computation, show that 2 < 2 0 41 + x3 dx 6.

The velocity of a car (in miles per hour) is given by v(t) = 40t 10t2, where t is in hours. (a) Write a definite integral for the distance the car travels during the first three hours. (b) Sketch a graph of velocity against time and represent the distance traveled during the first three hours as an area on your graph. (c) Use a computer or calculator to find this distance.

The worlds oil is being consumed at a continuously increasing rate, f(t) (in billions of barrels per year), where t is in years since the start of 1995. (a) Write a definite integral which represents the total quantity of oil used between the start of 1995 and the start of 2010. (b) Suppose r = 32(1.05)t. Using a left-hand sumwith five subdivisions, find an approximate value for the total quantity of oil used between the start of 1995 and the start of 2010. (c) Interpret each of the five terms in the sum from part (b) in terms of oil consumption.

A car moves along a straight line with velocity, in feet/second, given by v = 6 2t for t 0. (a) With forward being the direction of positive velocity, when is the car moving forward? Backward? Stopped? (b) Does the car ever return to its starting point? If so, when? (c) With position measured from its starting point, when is the car farthest forward? Farthest backward?

The marginal cost function of a product, in dollars per unit, is C(q) = q2 50q +700. If fixed costs are $500, find the total cost to produce 50 items.

A warehouse charges its customers $5 per day for every 10 cubic feet of space used for storage. Figure 5.87 records the storage used by one company over a month. How much will the company have to pay? 10 20 30 10,000 20,000 30,000 days cubic feet Figure 5.87

One of the earliest pollution problems brought to the attention of the Environmental Protection Agency (EPA) was the case of Sioux Lake in eastern South Dakota. For years a small paper plant located nearby had been discharging waste containing carbon tetrachloride (CCl4) into the waters of the lake. At the time the EPA learned of the situation, the chemical was entering at a rate of 16 cubic yards/year. The agency immediately ordered the installation of filters designed to slow (and eventually stop) the flow of CCl4 from the mill. Implementation of this program took exactly three years, during which the flow of pollutant was steady at 16 cubic yards/year. Once the filters were installed, the flow declined. Fromthe time the filters were installed until the time the flow stopped, the rate of flow was well approximated by Rate (in cubic yards/year) = t2 14t + 49, where t is time measured in years since the EPA learned of the situation (thus, t 3). (a) Draw a graph showing the rate of CCl4 flow into the lake as a function of time, beginning at the time the EPA first learned of the situation. 280 Chapter Five ACCUMULATED CHANGE: THE DEFINITE INTEGRAL (b) How many years elapsed between the time the EPA learned of the situation and the time the pollution flow stopped entirely? (c) How much CCl4 entered the waters during the time shown in the graph in part (a)?

(a) Graph x3 5x2 +4x, marking x = 1, 2, 3, 4, 5. (b) Use your graph and the area interpretation of the definite integral to decide which of the five numbers In = < n 0 (x35x2+4x) dx for n = 1, 2, 3, 4, 5 is largest. Which is smallest? How many of the numbers are positive? (Dont calculate the integrals.)

A mouse moves back and forth in a straight tunnel, attracted to bits of cheddar cheese alternately introduced to and removed from the ends (right and left) of the tunnel. The graph of the mouses velocity, v, is given in Figure 5.88, with positive velocity corresponding to motion toward the right end. Assuming that the mouse starts (t = 0) at the center of the tunnel, use the graph to estimate the time(s) at which: (a) The mouse changes direction. (b) The mouse is moving most rapidly to the right; to the left. (c) The mouse is farthest to the right of center; farthest to the left. (d) The mouses speed (i.e., the magnitude of its velocity) is decreasing. (e) The mouse is at the center of the tunnel. 30 20 10 0 10 20 30 v (cm/sec) 5 10 15 20 25 30 35 40 t (sec) Figure 5.88

Pollution is being dumped into a lake at a rate which is increasing at a constant rate from 10 kg/year to 50 kg/year until a total of 270 kg has been dumped. Sketch a graph of the rate at which pollution is being dumped in the lake against time. How long does it take until 270 kg has been dumped?

Calculate F(b) for b = 0, 0.5, 1, 1.5, 2, 2.5.

Using a graph of F, decide where F is increasing and where F is decreasing for 0 x 2.5.

Does F have a maximum value for 0 x 2.5? If so, what is it, and at what value of x does it occur?

What is the average value of the function f in Figure 5.89 over the interval 1 x 6? 1 2 3 4 5 6 1 2 3 f(x) x Figure 5.89

Find the average value of g(t) = et over the interval 0 t 10.

Find the average value of the function f(x) = 5+4xx2 between x = 0 and x = 3.

A service station orders 100 cases of motor oil every 6 months. The number of cases of oil remaining t months after the order arrives is modeled by f(t) = 100e0.5t . (a) How many cases are there at the start of the sixmonth period? How many cases are left at the end of the six-month period? (b) Find the average number of cases in inventory over the six-month period.

The quantity of a radioactive substance at time t is Q(t) = 4(0.96)t grams. (a) Find Q(10) and Q(20). (b) Find the average of Q(10) and Q(20). (c) Find the average value of Q(t) over the interval 10 t 20. (d) Use the graph of Q(t) to explain the relative sizes of your answers in parts (b) and (c).

a b 2 6 10 f(x) x

a b 25 50 75 100 F

The function f in Figure 5.90 is symmetric about the yaxis. Consider the average value of f over the following intervals: I. 0 x 1 II. 0 x 2 III. 0 x 5 IV. 2 x 2 (a) For which interval is the average value of f least? (b) For which interval is the average value of f greatest? (c) For which pair of intervals are the average values equal?

For the function f in Figure 5.90, write an expression involving one or more definite integrals of f that denote: (a) The average value of f for 0 x 5. (b) The average value of |f| for 0 x 5.