Ab Calculus Integrals as Net Change and Volume Review Answers

Learning Objectives

• five.4.1 Apply the basic integration formulas.
• 5.4.2 Explain the significance of the cyberspace change theorem.
• 5.4.3 Use the net change theorem to solve applied problems.
• 5.iv.4 Utilize the integrals of odd and even functions.

In this department, we utilize some basic integration formulas studied previously to solve some key applied problems. It is important to note that these formulas are presented in terms of indefinite integrals. Although definite and indefinite integrals are closely related, there are some key differences to continue in listen. A definite integral is either a number (when the limits of integration are constants) or a single function (when one or both of the limits of integration are variables). An indefinite integral represents a family of functions, all of which differ by a abiding. As you become more familiar with integration, you will go a experience for when to apply definite integrals and when to use indefinite integrals. You will naturally select the right approach for a given problem without thinking too much virtually it. However, until these concepts are cemented in your mind, think carefully most whether you need a definite integral or an indefinite integral and make sure you lot are using the proper notation based on your choice.

Basic Integration Formulas

Recall the integration formulas given in the table in Antiderivatives and the rule on properties of definite integrals. Let'south await at a few examples of how to apply these rules.

Example 5.23

Integrating a Function Using the Ability Rule

Use the power rule to integrate the function ${\displaystyle {\int }_1^{\mathrm{iv}}\sqrt{t}\left(1+t\right)dt}.$

Checkpoint 5.21

Find the definite integral of $f\left(x\right)={\mathrm{ten}}^{\mathrm{ii}}-3x$ over the interval $\left[\mathrm{i},3\right].$

The Cyberspace Modify Theorem

The net change theorem considers the integral of a charge per unit of change . It says that when a quantity changes, the new value equals the initial value plus the integral of the rate of alter of that quantity. The formula can exist expressed in two ways. The second is more familiar; information technology is simply the definite integral.

Theorem 5.6

Cyberspace Change Theorem

The new value of a irresolute quantity equals the initial value plus the integral of the rate of change:

$$\begin{array}{}\\ \\ F\left(b\right)=F\left(a\right)+{\displaystyle {\int }_a^{b}F\prime \left(x\right)dx}\hfill \\ \hfill \text{or}\hfill \\ {\displaystyle {\int }_a^{b}F\prime \left(\mathrm{10}\right)dx=F\left(b\right)-F\left(a\right)}.\hfill \end{array}$$

(5.eighteen)

Subtracting $F\left(a\right)$ from both sides of the first equation yields the second equation. Since they are equivalent formulas, which one we utilise depends on the application.

The significance of the cyberspace change theorem lies in the results. Net change can be applied to expanse, distance, and volume, to name merely a few applications. Net change accounts for negative quantities automatically without having to write more one integral. To illustrate, let'southward apply the net modify theorem to a velocity function in which the upshot is displacement.

We looked at a unproblematic example of this in The Definite Integral. Suppose a auto is moving due north (the positive management) at xl mph betwixt ii p.chiliad. and four p.m., and then the motorcar moves s at 30 mph between iv p.m. and 5 p.m. We tin graph this move as shown in Effigy 5.32.

Figure 5.32 The graph shows speed versus fourth dimension for the given motility of a car.

Just equally nosotros did before, we can utilize definite integrals to calculate the cyberspace deportation besides as the full altitude traveled. The internet deportation is given by

$$\begin{array}{cc}{\displaystyle {\int }_2^{5}v}(t)dt\hfill & ={\displaystyle {\int }_2^{4}4}0dt+{\displaystyle {\int }_4^5}\mathrm{-thirty}dt\hfill \\ & =80-30\hfill \\ & =50.\hfill \end{array}$$

Thus, at 5 p.m. the car is 50 mi north of its starting position. The total altitude traveled is given by

$$\begin{array}{}\\ \\ {{\displaystyle \int }}_2^5\left|5(t)\right|dt\hfill & ={\displaystyle {\int }_2^{4}4}0dt+{{\displaystyle \int }}_{\mathrm{iv}}^{5}30dt\hfill \\ & =80+30\hfill \\ & =110.\hfill \end{array}$$

Therefore, between 2 p.one thousand. and 5 p.m., the automobile traveled a full of 110 mi.

To summarize, cyberspace displacement may include both positive and negative values. In other words, the velocity role accounts for both frontward altitude and astern distance. To find net displacement, integrate the velocity role over the interval. Full altitude traveled, on the other hand, is ever positive. To find the total distance traveled by an object, regardless of direction, nosotros demand to integrate the absolute value of the velocity part.

Example five.24

Finding Net Displacement

Given a velocity function $5\left(t\right)=3t-\mathrm{five}$ (in meters per second) for a particle in motility from time $t=0$ to fourth dimension $t=3,$ find the cyberspace displacement of the particle.

Example five.25

Finding the Full Distance Traveled

Use Instance 5.24 to find the total altitude traveled past a particle co-ordinate to the velocity part $\mathrm{five}\left(t\right)=3t-\mathrm{five}$ g/sec over a fourth dimension interval $\left[0,3\right].$

Checkpoint 5.22

Find the cyberspace displacement and total distance traveled in meters given the velocity function $f\left(t\right)=\frac{1}{2}{\mathrm{due\; east}}^t-2$ over the interval $\left[0,\mathrm{two}\right].$

Applying the Net Change Theorem

The net change theorem can be applied to the menstruum and consumption of fluids, as shown in Instance 5.26.

Instance 5.26

How Many Gallons of Gasoline Are Consumed?

If the motor on a motorboat is started at $t=0$ and the boat consumes gasoline at the charge per unit of $5-t^{\mathrm{iii}}$ gal/hr, how much gasoline is used in the first 2 hours?

Example five.27

Chapter Opener: Iceboats

Figure five.34 (credit: modification of piece of work by Carter Brown, Flickr)

Every bit nosotros saw at the beginning of the chapter, top iceboat racers (Figure 5.one) can reach speeds of up to five times the wind speed. Andrew is an intermediate iceboater, though, and then he attains speeds equal to just twice the wind speed. Suppose Andrew takes his iceboat out i morning when a low-cal 5-mph breeze has been blowing all morning. Every bit Andrew gets his iceboat set upwardly, though, the wind begins to pick up. During his first half hour of iceboating, the wind speed increases according to the function $v\left(t\right)=\mathrm{xx}t+5.$ For the second half hour of Andrew'due south outing, the wind remains steady at 15 mph. In other words, the wind speed is given by

$$v\left(t\right)=\{\begin{array}{lll}20t+\mathrm{five}\hfill & \text{for}\hfill & 0\le t\le \frac{1}{\mathrm{ii}}\hfill \\ \mathrm{fifteen}\hfill & \text{for}\hfill & \frac{1}{2}\le t\le 1.\hfill \end{array}$$

Recalling that Andrew's iceboat travels at twice the wind speed, and assuming he moves in a straight line away from his starting point, how far is Andrew from his starting point after 1 hr?

Checkpoint v.23

Suppose that, instead of remaining steady during the second half hour of Andrew's outing, the wind starts to die downward according to the function $v\left(t\right)=\mathrm{-10}t+15.$ In other words, the wind speed is given by

$$\mathrm{five}\left(t\right)=\{\begin{array}{lll}\mathrm{twenty}t+5\hfill & \text{for}\hfill & 0\le t\le \frac{1}{2}\hfill \\ -10t+15\hfill & \text{for}\hfill & \frac{1}{2}\le t\le \mathrm{one}.\hfill \end{array}$$

Nether these conditions, how far from his starting point is Andrew after ane 60 minutes?

Integrating Even and Odd Functions

Nosotros saw in Functions and Graphs that an even function is a function in which $f\left(\text{−}\mathrm{10}\right)=f\left(\mathrm{ten}\right)$ for all x in the domain—that is, the graph of the curve is unchanged when x is replaced with −10. The graphs of fifty-fifty functions are symmetric nearly the y-axis. An odd function is 1 in which $f\left(\text{−}x\right)=\text{−}f\left(x\right)$ for all x in the domain, and the graph of the part is symmetric about the origin.

Integrals of even functions, when the limits of integration are from −a to a, involve two equal areas, because they are symmetric about the y-axis. Integrals of odd functions, when the limits of integration are similarly $\left[\text{−}a,a\right],$ evaluate to zero because the areas in a higher place and below the x-centrality are equal.

Rule: Integrals of Even and Odd Functions

For continuous fifty-fifty functions such that $f\left(\text{−}\mathrm{10}\right)=f\left(x\right),$

$${\displaystyle {\int }_{\text{−}a}^{a}f\left(\mathrm{10}\right)dx=2{\displaystyle {\int }_0^{a}f\left(x\right)dx}}.$$

For continuous odd functions such that $f\left(\text{−}\mathrm{10}\right)=\text{−}f\left(\mathrm{10}\right),$

$${\displaystyle {\int }_{\text{−}a}^{a}f\left(\mathrm{ten}\right)d\mathrm{ten}=0}.$$

Instance 5.28

Integrating an Even Part

Integrate the even function ${\displaystyle {\int }_{\mathrm{-2}}^{\mathrm{ii}}\left(3{\mathrm{10}}^8-2\right)dx}$ and verify that the integration formula for even functions holds.

Example 5.29

Integrating an Odd Function

Evaluate the definite integral of the odd office $\mathrm{-five}\text{sin}x$ over the interval $\left[\text{−}\pi ,\pi \right].$

Checkpoint five.24

Integrate the function ${\displaystyle {\int }_{\mathrm{-2}}^2{x}^{4}d\mathrm{ten}}.$

Department v.four Exercises

Utilise basic integration formulas to compute the following antiderivatives or definite integrals.

207.

${\displaystyle \int \left(\sqrt{\mathrm{10}}-\frac{1}{\sqrt{x}}\right)dx}$

208 .

${\displaystyle \int \left(e^{2\mathrm{10}}-\frac{\mathrm{ane}}{\mathrm{ii}}{e}^{x\text{/}2}\right)d\mathrm{10}}$

210 .

${\displaystyle \int \frac{x-\mathrm{i}}{{\mathrm{ten}}^2}dx}$

211.

${\displaystyle {\int }_0^{\pi }\left(\text{sin}x-\text{cos}\mathrm{10}\right)d\mathrm{10}}$

212 .

${\displaystyle {\int }_0^{\pi \text{/}\mathrm{ii}}\left(x-\text{sin}x\right)d\mathrm{10}}$

213.

Write an integral that expresses the increase in the perimeter $P\left(s\right)$ of a square when its side length southward increases from ii units to 4 units and evaluate the integral.

214 .

Write an integral that quantifies the change in the expanse $A\left(\mathrm{due\; south}\right)=s^2$ of a square when the side length doubles from Southward units to 2S units and evaluate the integral.

215.

A regular North-gon (an Due north-sided polygon with sides that have equal length s, such as a pentagon or hexagon) has perimeter Ns. Write an integral that expresses the increase in perimeter of a regular Northward-gon when the length of each side increases from 1 unit to 2 units and evaluate the integral.

216 .

The area of a regular pentagon with side length $a>0$ is pa ² with $p=\frac{\mathrm{ane}}{\mathrm{four}}\sqrt{5+\sqrt{\mathrm{five}+2\sqrt{\mathrm{v}}}}.$ The Pentagon in Washington, DC, has inner sides of length 360 ft and outer sides of length 920 ft. Write an integral to express the area of the roof of the Pentagon co-ordinate to these dimensions and evaluate this surface area.

217.

A dodecahedron is a Platonic solid with a surface that consists of 12 pentagons, each of equal area. Past how much does the surface expanse of a dodecahedron increase as the side length of each pentagon doubles from 1 unit to 2 units?

218 .

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

219.

Write an integral that quantifies the change in the area of the surface of a cube when its side length doubles from s unit to iidue south units and evaluate the integral.

220 .

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.

221.

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

222 .

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

223.

Suppose that a particle moves forth a straight line with velocity $v\left(t\right)=4-\mathrm{ii}t,$ where $0\le t\le \mathrm{two}$ (in meters per second). Detect the deportation at time t and the total distance traveled up to $t=2.$

224 .

Suppose that a particle moves along a direct line with velocity defined by $\mathrm{five}\left(t\right)=t^2-\mathrm{three}t-\mathrm{eighteen},$ where $0\le t\le 6$ (in meters per second). Discover the displacement at time t and the total distance traveled up to $t=6.$

225.

Suppose that a particle moves forth a direct line with velocity defined by $\mathrm{five}\left(t\right)=\left|2t-\mathrm{half\; dozen}\right|,$ where $0\le t\le \mathrm{vi}$ (in meters per second). Find the displacement at time t and the total distance traveled up to $t=\mathrm{vi}.$

226 .

Suppose that a particle moves forth a straight line with acceleration defined by $a\left(t\right)=t-\mathrm{three},$ where $0\le t\le \mathrm{half-dozen}$ (in meters per 2nd). Find the velocity and displacement at time t and the full altitude traveled upward to $t=6$ if $v\left(0\right)=\mathrm{three}$ and $d\left(0\right)=0.$

227.

A ball is thrown upwardly from a height of ane.5 thou at an initial speed of 40 m/sec. Acceleration resulting from gravity is −9.8 m/secⁱⁱ. Neglecting air resistance, solve for the velocity $\mathrm{five}\left(t\right)$ and the meridian $h\left(t\right)$ of the ball t seconds after it is thrown and before it returns to the ground.

228 .

A brawl is thrown upward from a top of 3 m at an initial speed of lx m/sec. Acceleration resulting from gravity is −ix.8 m/sec². Neglecting air resistance, solve for the velocity $v\left(t\right)$ and the height $h\left(t\right)$ of the ball t seconds later it is thrown and before it returns to the ground.

229.

The surface area $A\left(t\right)$ of a round shape is growing at a constant rate. If the area increases from fourπ units to 9π units betwixt times $t=\mathrm{two}$ and $t=3,$ find the net change in the radius during that time.

230 .

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

231.

H2o flows into a conical tank with cross-sectional area πx ² at summit x and volume $\frac{\pi {\mathrm{ten}}^3}{3}$ up to summit ten. If water flows into the tank at a rate of 1 mⁱⁱⁱ/min, find the height of h2o in the tank after 5 min. Find the change in top betwixt 5 min and 10 min.

232 .

A horizontal cylindrical tank has cross-sectional area $A\left(x\right)=4\left(6x-x^{\mathrm{ii}}\right){\mathrm{1000}}^2$ at height 10 meters above the bottom when $x\le 3.$

1. The volume 5 between heights a and b is ${\displaystyle {\int }_a^{b}A\left(x\right)dx}.$ Discover the volume at heights between 2 1000 and 3 m.
2. Suppose that oil is being pumped into the tank at a rate of fifty Fifty/min. Using the concatenation rule, $\frac{d\mathrm{10}}{dt}=\frac{dx}{dV}\frac{dV}{dt},$ at how many meters per infinitesimal is the height of oil in the tank changing, expressed in terms of x, when the height is at x meters?
3. How long does it accept to fill the tank to 3 grand starting from a fill level of 2 m?

233.

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

Hour	Power	Hour	Power
1	28	13	48
2	25	14	49
3	24	15	49
four	23	16	fifty
5	24	17	l
6	27	18	50
7	29	19	46
viii	32	20	43
9	34	21	42
10	39	22	40
11	42	23	37
12	46	24	34

Find the total amount of free energy in gigawatt-hours (gW-h) consumed past the urban center in a typical 24-hr period.

234 .

The average residential electrical ability apply (in hundreds of watts) per hour is given in the post-obit table.

Hr	Power	60 minutes	Ability
1	viii	13	12
2	6	14	xiii
3	v	15	14
four	four	16	15
5	5	17	17
6	6	eighteen	xix
7	seven	nineteen	18
8	8	20	17
nine	9	21	16
ten	x	22	sixteen
11	10	23	13
12	11	24	xi

1. Compute the average total energy used in a twenty-four hour period in kilowatt-hours (kWh).
2. If a ton of coal generates 1842 kWh, how long does it take for an boilerplate residence to burn down a ton of coal?
3. Explain why the data might fit a plot of the form $p\left(t\right)=11.5-\mathrm{vii.v}\text{sin}\left(\frac{\pi t}{12}\right).$

235.

The data in the following tabular array are used to estimate the boilerplate power output produced past Peter Sagan for each of the last 18 sec of Phase 1 of the 2012 Bout de France.

2nd	Watts	2nd	Watts
i	600	x	1200
2	500	xi	1170
3	575	12	1125
four	1050	13	1100
5	925	xiv	1075
6	950	15	1000
7	1050	16	950
8	950	17	900
9	1100	18	780

Table 5.half-dozen Average Power Output Source: sportsexercisengineering.com

Approximate the cyberspace energy used in kilojoules (kJ), noting that 1W = 1 J/due south, and the average power output past Sagan during this time interval.

236 .

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.

Minutes	Watts	Minutes	Watts
15	200	165	170
30	180	180	220
45	190	195	140
threescore	230	210	225
75	240	225	170
90	210	240	210
105	210	255	200
120	220	270	220
135	210	285	250
150	150	300	400

Table 5.7 Average Ability Output Source: sportsexercisengineering.com

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

237.

The distribution of incomes equally of 2012 in the United states in $5000 increments is given in the following table. The chiliadthursday row denotes the percentage of households with incomes between $\mathrm{\$5000}x\mathrm{1000}$ and $5000\mathrm{10}\mathrm{one\; thousand}+4999.$ The row $\mathrm{thou}=40$ contains all households with income between $200,000 and $250,000.

0	3.v	21	1.v
one	iv.i	22	1.4
2	5.9	23	1.3
3	five.vii	24	ane.3
4	v.ix	25	1.1
v	v.4	26	1.0
6	5.five	27	0.75
vii	5.one	28	0.eight
8	iv.viii	29	ane.0
nine	four.ane	30	0.6
ten	4.3	31	0.6
eleven	three.v	32	0.5
12	3.7	33	0.five
13	3.2	34	0.4
xiv	3.0	35	0.3
15	2.8	36	0.iii
16	2.5	37	0.iii
17	two.2	38	0.2
xviii	2.2	39	1.8
xix	1.8	40	2.three
20	two.1

Table v.8 Income Distributions Source: http://www.census.gov/prod/2013pubs/p60-245.pdf

1. Estimate the percent of U.South. households in 2012 with incomes less than $55,000.
2. What percent of households had incomes exceeding $85,000?
3. Plot the data and effort to fit its shape to that of a graph of the course $a\left(\mathrm{ten}+c\right){\mathrm{due\; east}}^{\text{−}b\left(\mathrm{10}+e\right)}$ for suitable $a,b,c.$

238 .

Newton's law of gravity states that the gravitational force exerted past an object of mass M and one of mass m with centers that are separated by a distance r is $F=M\frac{mM}{r^2},$ with G an empirical constant $\mathrm{1000}=6.67x{\mathrm{ten}}^{\mathrm{-eleven}}{m}^{\mathrm{three}}\text{/}\left(m\mathrm{one\; thousand}·{\mathrm{south}}^2\right).$ The work done by a variable strength over an interval $\left[a,b\right]$ is divers every bit $W={\displaystyle {\int }_a^{b}F\left(x\right)dx}.$ If Earth has mass $\mathrm{five.97219}\times {10}^{24}$ and radius 6371 km, compute the amount of work to elevate a polar weather satellite of mass 1400 kg to its orbiting distance of 850 km above Earth.

239.

For a given motor vehicle, the maximum achievable deceleration from braking is approximately seven m/sec^two on dry concrete. On moisture asphalt, it is approximately 2.5 m/sec². Given that i mph corresponds to 0.447 m/sec, detect the total distance that a car travels in meters on dry concrete later the brakes are applied until it comes to a complete cease if the initial velocity is 67 mph (thirty thou/sec) or if the initial braking velocity is 56 mph (25 m/sec). Find the respective distances if the surface is glace wet asphalt.

240 .

John is a 25-twelvemonth onetime man who weighs 160 lb. He burns $500-\mathrm{fifty}t$ calories/hr while riding his bike for t hours. If an oatmeal cookie has 55 cal and John eats fourt cookies during the tth hr, how many net calories has he lost subsequently 3 hours riding his wheel?

241.

Sandra is a 25-year old adult female who weighs 120 lb. She burns $300-50t$ cal/hour while walking on her treadmill. Her caloric intake from drinking Gatorade is 100t calories during the tth 60 minutes. What is her net decrease in calories after walking for 3 hours?

242 .

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 charge per unit of 0.4 mpg/mph betwixt 50 mph and lxxx mph. What is the efficiency in miles per gallon if the auto is cruising at 50 mph? What is the efficiency in miles per gallon if the car is cruising at eighty mph? If gasoline costs $iii.50/gal, what is the cost of fuel to bulldoze 50 mi at 40 mph, at 50 mph, and at 80 mph?

243.

Although some engines are more than efficient at given a horsepower than others, on average, fuel efficiency decreases with horsepower at a rate of $\mathrm{one}\text{/}25$ 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?

244 .

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

Taxable Income Range	The Tax Is …	… Of the Amount Over
$0–$8925	10%	$0
$8925–$36,250	$892.50 + 15%	$8925
$36,250–$87,850	$4,991.25 + 25%	$36,250
$87,850–$183,250	$17,891.25 + 28%	$87,850
$183,250–$398,350	$44,603.25 + 33%	$183,250
$398,350–$400,000	$115,586.25 + 35%	$398,350
> $400,000	$116,163.75 + 39.6%	$400,000

Table five.9 Federal Income Revenue enhancement Versus Taxable Income Source: http://www.irs.gov/pub/irs-prior/i1040tt--2013.pdf.

Suppose that Steve just received a $10,000 enhance. How much of this enhance is left later on federal taxes if Steve'due south salary before receiving the raise was $40,000? If it was $ninety,000? If it was $385,000?

245.

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

Highway Speed Range (mph)	Vehicles per 60 minutes per Lane	Density Range (vehicles/mi)
> 60	< 600	< 10
60–57	600–thou	10–twenty
57–54	1000–1500	20–30
54–46	1500–1900	30–45
46–30	1900–2100	45–lxx
<xxx	Unstable	70–200

Table 5.x

1. Plot vehicles per hour per lane on the x-axis and highway speed on the y-centrality.
2. Compute the average subtract in speed (in miles per hr) per unit increase in congestion (vehicles per hour per lane) as the latter increases from 600 to thou, 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?
3. Plot minutes per mile (threescore times the reciprocal of miles per hour) as a office of vehicles per hour per lane. Is this part linear?

For the next two exercises use the data in the following tabular array, which displays baldheaded eagle populations from 1963 to 2000 in the continental United States.

Twelvemonth	Population of Breeding Pairs of Baldheaded Eagles
1963	487
1974	791
1981	1188
1986	1875
1992	3749
1996	5094
2000	6471

Table 5.11 Population of Breeding Bald Hawkeye Pairs Source: http://world wide web.fws.gov/Midwest/eagle/population/chtofprs.html.

246 .

[T] The graph below plots the quadratic $p\left(t\right)=\mathrm{vi.48}{t}^{\mathrm{two}}-\mathrm{eighty.3}1t+585.69$ against the data in preceding table, normalized so that $t=0$ corresponds to 1963. Guess the average number of bald eagles per year nowadays for the 37 years by computing the boilerplate value of p over $\left[0,37\right].$

247.

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

248 .

[T] Suppose you get on a route trip and record your speed at every half 60 minutes, equally compiled in the post-obit table. The best quadratic fit to the information is $q\left(t\right)=5{\mathrm{ten}}^{\mathrm{two}}-\mathrm{xi}x+49\text{,}$ shown in the accompanying graph. Integrate q to estimate the total altitude driven over the 3 hours.

Time (hr)	Speed (mph)
0 (get-go)	fifty
ane	40
2	l
3	threescore

Equally a machine accelerates, it does non advance at a constant rate; rather, the acceleration is variable. For the following exercises, use the following table, which contains the acceleration measured at every second as a driver merges onto a pike.

Time (sec)	Acceleration (mph/sec)
1	eleven.2
ii	10.6
three	8.1
iv	5.4
5	0

249.

[T] The accompanying graph plots the all-time quadratic fit, $a\left(t\right)=\mathrm{-0.seventy}{t}^2+\mathrm{ane.44}t+10.44,$ to the information from the preceding table. Compute the average value of $a\left(t\right)$ to estimate the average acceleration between $t=0$ and $t=5.$

250 .

[T] Using your acceleration equation from the previous exercise, detect the respective velocity equation. Bold the final velocity is 0 mph, find the velocity at time $t=0.$

251.

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

252 .

[T] The number of hamburgers sold at a eating place throughout the twenty-four hours is given in the following tabular array, with the accompanying graph plotting the best cubic fit to the data, $b\left(t\right)=0.12t^{\mathrm{iii}}-\mathrm{2.xiii}{t}^3+12.13t+3.91,$ with $t=0$ corresponding to ix a.thou. and $t=12$ corresponding to nine p.chiliad. Compute the average value of $b\left(t\right)$ to approximate the average number of hamburgers sold per 60 minutes.

Hours Past Midnight	No. of Burgers Sold
9	3
12	28
15	20
eighteen	thirty
21	45

253.

[T] An athlete runs by a motion detector, which records her speed, as displayed in the following table. The best linear fit to this data, $\ell \left(t\right)=\mathrm{-0.068}t+\mathrm{5.fourteen}\text{,}$ is shown in the accompanying graph. Use the boilerplate value of $\ell (t)$ between $t=0$ and $t=40$ to approximate the runner'due south average speed.

Minutes	Speed (chiliad/sec)
0	5
10	4.eight
20	three.6
thirty	iii.0
forty	two.5

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Source: https://openstax.org/books/calculus-volume-1/pages/5-4-integration-formulas-and-the-net-change-theorem

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