Chapter 5

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

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

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 2\(a\) units?

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 2 s units and evaluate the integral.

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

Suppose that a particle moves along a straight line with velocity \(v(t)=4-2 t,\) where \(0 \leq t \leq 2\) (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 \(v(t)=t^{2}-3 t-18, \quad\) where \(0 \leq t \leq 6\) (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 \(v(t)=|2 t-6|, \quad\) where \(0 \leq t \leq 6\) (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 \(a(t)=t-3, \quad\) where \(0 \leq t \leq 6\) (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 \(\mathrm{m}\) at an initial speed of 40 \(\mathrm{m} / \mathrm{sec}\) . Acceleration resulting from gravity is \(-9.8 \mathrm{m} / \mathrm{sec}^{2} .\) 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 \(3 \mathrm{~m}\) at an initial speed of \(60 \mathrm{~m} / \mathrm{sec}\). Acceleration resulting from gravity is \(-9.8 \mathrm{~m} / \mathrm{sec}^{2}\). 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.

The area \(A(t)\) of a circular shape is growing at a constant rate. If the area increases from 4\(\pi\) units to 9\(\pi\) units between times \(t=2\) and \(t=3,\) find the net change in the radius during that time.

A spherical balloon is being inflated at a constant rate. If the volume of the balloon changes from \(36 \pi\) in. \(^{3}\) to \(288 \pi\) 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 \(\pi x^{2}\) at height \(x\) and volume \(\frac{\pi x^{3}}{3}\) up to height \(x\). If water flows into the tank at a rate of \(1 \mathrm{~m}^{3} / \mathrm{min}\), 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 \(A(x)=4\left(6 x-x^{2}\right) m^{2}\) at height \(x\) meters above the bottom when \(x \leq 3\) . a. The volume \(V\) between heights \(a\) and \(b\) is \(\int_{a}^{b} A(x) d x\) . Find the volume at heights between 2 m and 3 \(\mathrm{m}\) . b. Suppose that oil is being pumped into the tank \(\frac{d x}{d t}=\frac{d x}{d V} \frac{d V}{d t},\) 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 \(\mathrm{m}\) starting from a fill level of 2 \(\mathrm{m} ?\)

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 power in gigawatt-hours (gW-h) consumed by the city in a typical 24 -hour period.

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 \(F=G \frac{m M}{r^{2}}, \quad\) with \(G\) an empirical constant \(G=6.67 x 10^{-11} \mathrm{m}^{3} /\left(\mathrm{kg} \cdot \mathrm{s}^{2}\right) .\) The work done by a variable force over an interval \([a, b]\) is defined as \(W=\int_{a}^{b} F(x) d x\) . If Earth has mass \(5.97219 \times 10^{24}\) and radius 6371 \(\mathrm{km}\) , compute the amount of work to elevate a polar weather satellite of mass 1400 \(\mathrm{kg}\) to its orbiting altitude of 850 \(\mathrm{km}\) above Earth.

For a given motor vehicle, the maximum achievable deceleration from braking is approximately 7 \(\mathrm{m} / \mathrm{sec}^{2}\) on dry concrete. On wet asphalt, it is approximately 2.5 \(\mathrm{m} / \mathrm{sec}^{2}\) Given that 1 \(\mathrm{mph}\) corresponds to \(0.447 \mathrm{m} / \mathrm{sec},\) find the total distance that a car travels in meters on dry concrete after the brakes are applied until it comes to a complete stop if the initial velocity is 67 \(\mathrm{mph}(30 \mathrm{m} / \mathrm{sec})\) or if the initial braking velocity is 56 \(\mathrm{mph}(25 \mathrm{m} / \mathrm{sec}) .\) Find the corresponding distances if the surface is slippery wet asphalt.

John is a 25 -year old man who weighs 160 lb. He burns \(500-50 t\) calories/hr while riding his bike for \(t\) hours. If an oatmeal cookie has 55 cal and John eats 4\(t\) cookies during the th hour, how many net calories has he lost after 3 hours riding his bike?

John is a 25-year old man who weighs \(160 \mathrm{lb}\). He burns \(500-50 t\) calories/hr while riding his bike for \(t\) hours. If an oatmeal cookie has 55 cal and John eats \(4 t\) cookies 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 \(\mathrm{lb}\) . She burns \(300-50 t\) calhr while walking on her treadmill. Her caloric intake from drinking Gatorade is 100\(t\) calories during the tth hour. What is her net decrease in calories after walking for 3 hours?

Sandra is a 25-year old woman who weighs 120 lb. She burns \(300-50 t\) cal/hr while walking on her treadmill. Her caloric intake from drinking Gatorade is \(100 t\) calories during the th hour. What is her net decrease in calories after walking for 3 hours?

A motor vehicle has a maximum efficiency of 33 \(\mathrm{mpg}\) at a cruising speed of 40 \(\mathrm{mph}\) . The efficiency drops at a rate of 0.1 \(\mathrm{mph}\) hoh between 40 \(\mathrm{mph}\) and 50 \(\mathrm{mph}\) , and at a rate of 0.4 \(\mathrm{mpg} / \mathrm{mph}\) between 50 \(\mathrm{mph}\) and 80 \(\mathrm{mph}\) . 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 \(\mathrm{mph}\) ? If gasoline costs \(\$ 3.50 / \mathrm{gal}\) , what is the cost of fuel to drive 50 \(\mathrm{mi}\) at 40 \(\mathrm{mph}\) , at 50 \(\mathrm{mph}\) , and at 80 \(\mathrm{mph} ?\)

Although some engines are more efficient at given horsepower than others, on average, fuel efficiency decreases with horsepower at a rate of 1\(/ 25\) mpg/ horsepower. If a typical 50 -horsepower engine has an average fuel efficiency of 32 \(\mathrm{mpg}\) , what is the average fuel efficiency of an engine with the following horsepower: 150 , \(300,450 ?\)

\(\begin{array}{ll}& \text { [T] The graph below plots the quadratic }\end{array}\) \(p(t)=6.48 t^{2}-80.31 t+585.69\) against the data in preceding table, normalized so that \(t=0\) corresponds to 1963\. Estimate the average number of bald eagles per year present for the 37 years by computing the average value of \(p\) over [0,37]

ITI The graph below plots the cubic \(p(t)=0.07 t^{3}+2.42 t^{2}-25.63 t+521.23 \quad\) against the data in the preceding table, normalized so that \(t=0\) corresponds to 1963 . Estimate the average number of bald eagles per year present for the 37 years by computing the average value of \(p\) over \([0,37]\) .

[T] Suppose you go on a road trip and record your speed at every half hour, as compiled in the following table. The best quadratic fit to the data is \(q(t)=5 x^{2}-11 x+49,\) shown in the accompanying graph. Integrate \(q\) to estimate the total distance driven over the 3 hours.

ITI The accompanying graph plots the best quadratic fit, \(a(t)=-0.70 t^{2}+1.44 t+10.44,\) to the data from the preceding table. Compute the average value of \(a(t)\) to estimate the average acceleration between \(t=0\) and \(t=5\)

[T] The accompanying graph plots the best quadratic fit, \(a(t)=-0.70 t^{2}+1.44 t+10.44,\) to the data from the preceding table. Compute the average value of \(a(t)\) to estimate the average acceleration between \(t=0\) and \(t=5\)

[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, \(\ell(t)=-0.068 t+5.14,\) is shown in the accompanying graph. Use the average value of \(\ell(t)\) between \(t=0\) and \(t=40\) to estimate the runner's average speed.

Why is \(u\) -substitution referred to as change of variable?

If \(f=g \circ h,\) when reversing the chain rule, \(\frac{d}{d x}(g \circ h)(x)=g^{\prime}(h(x)) h^{\prime}(x), \quad\) should you take \(u=g(x)\) or \(u=h(x) ?\)

In the following exercises, verify each identity using differentiation. Then, using the indicated \(u\) -substitution, identify \(f\) such that the integral takes the form \(\int f(u) d u .\) $$ \int x \sqrt{x+1} d x=\frac{2}{15}(x+1)^{3 / 2}(3 x-2)+C ; u=x+1 $$

In the following exercises, verify each identity using differentiation. Then, using the indicated \(u\) -substitution, identify \(f\) such that the integral takes the form \(\int f(u) d u .\) $$ \int \frac{x^{2}}{\sqrt{x-1}} d x(x>1)=\frac{2}{15} \sqrt{x-1}\left(3 x^{2}+4 x+8\right)+C ; u=x-1 $$

In the following exercises, verify each identity using differentiation. Then, using the indicated \(u\) -substitution, identify \(f\) such that the integral takes the form \(\int f(u) d u .\) $$ \int x \sqrt{4 x^{2}+9} d x=\frac{1}{12}\left(4 x^{2}+9\right)^{3 / 2}+C ; u=4 x^{2}+9 $$

In the following exercises, verify each identity using differentiation. Then, using the indicated \(u\) -substitution, identify \(f\) such that the integral takes the form \(\int f(u) d u .\) $$ \int \frac{x}{\sqrt{4 x^{2}+9}} d x=\frac{1}{4} \sqrt{4 x^{2}+9}+C ; u=4 x^{2}+9 $$

In the following exercises, verify each identity using differentiation. Then, using the indicated \(u\) -substitution, identify \(f\) such that the integral takes the form \(\int f(u) d u .\) $$ \int \frac{x}{\left(4 x^{2}+9\right)^{2}} d x=-\frac{1}{8\left(4 x^{2}+9\right)} ; u=4 x^{2}+9 $$

In the following exercises, find the antiderivative using the indicated substitution. $$ \int(x+1)^{4} d x ; u=x+1 $$

In the following exercises, find the antiderivative using the indicated substitution. $$ \int(x-1)^{5} d x ; u=x-1 $$

In the following exercises, find the antiderivative using the indicated substitution. $$ \int(2 x-3)^{-7} d x ; u=2 x-3 $$

In the following exercises, find the antiderivative using the indicated substitution. $$ \int(3 x-2)^{-11} d x ; u=3 x-2 $$

In the following exercises, find the antiderivative using the indicated substitution. $$ \int \frac{x}{\sqrt{x^{2}+1}} d x ; u=x^{2}+1 $$

In the following exercises, find the antiderivative using the indicated substitution. $$ \int \frac{x}{\sqrt{1-x^{2}}} d x ; u=1-x^{2} $$

In the following exercises, find the antiderivative using the indicated substitution. $$ \int(x-1)\left(x^{2}-2 x\right)^{3} d x ; u=x^{2}-2 x $$

In the following exercises, find the antiderivative using the indicated substitution. $$ \int \cos ^{3} \theta d \theta ; u=\sin \theta\left(\operatorname{Hint} \cos ^{2} \theta=1-\sin ^{2} \theta\right) $$

In the following exercises, find the antiderivative using the indicated substitution. $$ \int \sin ^{3} \theta d \theta ; u=\cos \theta\left(\operatorname{Hint} : \sin ^{2} \theta=1-\cos ^{2} \theta\right) $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int x(1-x)^{99} d x $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int t\left(1-t^{2}\right)^{10} d t $$