Chapter 5

Estimate \(\int_{0}^{6} 2^{x} d x\) using a left-hand sum with \(n=2\).

If the marginal cost function \(C^{\prime}(q)\) is measured in dollars per ton, and \(q\) gives the quantity in tons, what are the units of measurement for \(\int_{800}^{900} C^{\prime}(q) d q ?\) What does this integral represent?

The following table gives the emissions, \(E\), of nitrogen oxides in millions of metric tons per year in the US. Let \(t\) be the number of years since 1970 and \(E=f(t)\). (a) What are the units and meaning of \(\int_{0}^{30} f(t) d t\) ? (b) Estimate \(\int_{0}^{30} f(t) d t\). $$ \begin{array}{l|l|l|l|l|l|l|l} \hline \text { Year } & 1970 & 1975 & 1980 & 1985 & 1990 & 1995 & 2000 \\ \hline E & 26.9 & 26.4 & 27.1 & 25.8 & 25.5 & 25.0 & 22.6 \\ \hline \end{array} $$

Find the area under \(y=x^{3}+2\) between \(x=0\) and \(x=2\). Sketch this area.

Estimate \(\int_{0}^{12} \frac{1}{x+1} d x\) using a left-hand sum with \(n=3\).

The marginal cost function of a product, in dollars per unit, is \(C^{\prime}(q)=q^{2}-50 q+700\). If fixed costs are \(\$ 500\), find the total cost to produce 50 items.

Find the area under \(P=100(0.6)^{t}\) between \(t=0\) and \(t=8 .\)

Use the following table to estimate \(\int_{0}^{25} f(x) d x\). $$ \begin{array}{c|c|c|c|c|c|c} \hline x & 0 & 5 & 10 & 15 & 20 & 25 \\ \hline f(x) & 100 & 82 & 69 & 60 & 53 & 49 \\ \hline \end{array} $$

Explain in words what the integral represents and give units. \(\int_{1}^{3} v(t) d t\), where \(v(t)\) is velocity in meters/sec and \(t\) is time in seconds.

The total cost in dollars to produce \(q\) units of a product is \(C(q)\). Fixed costs are \(\$ 20,000\). The marginal cost is $$ C^{\prime}(q)=0.005 q^{2}-q+56 . $$ (a) On a graph of \(C^{\prime}(q)\), illustrate graphically the total variable cost of producing 150 units. (b) Estimate \(C(150)\), the total cost to produce 150 units. (c) Find the value of \(C^{\prime}(150)\) and interpret your answer in terms of costs of production. (d) Use parts (b) and (c) to estimate \(C(151)\).

Find the area between \(y=x+5\) and \(y=2 x+1\) between \(x=0\) and \(x=2\).

Use the following table to estimate \(\int_{3}^{4} W(t) d t\). What are \(n\) and \(\Delta t ?\) $$ \begin{array}{c|c|c|c|c|c|c} \hline t & 3.0 & 3.2 & 3.4 & 3.6 & 3.8 & 4.0 \\ \hline W(t) & 25 & 23 & 20 & 15 & 9 & 2 \\ \hline \end{array} $$

Explain in words what the integral represents and give units. \(\int_{0}^{6} a(t) d t\), where \(a(t)\) is acceleration in \(\mathrm{km} / \mathrm{hr}^{2}\) and \(t\) is time in hours.

Find the area enclosed by \(y-3 x\) and \(y=x^{2}\).

Use the following table to estimate \(\int_{0}^{15} f(x) d x\). $$ \begin{array}{c|c|c|c|c|c|c} \hline x & 0 & 3 & 6 & 9 & 12 & 15 \\ \hline f(x) & 50 & 48 & 44 & 36 & 24 & 8 \\ \hline \end{array} $$

Explain in words what the integral represents and give units. \(\int_{2000}^{2004} f(t) d t\), where \(f(t)\) is the rate at which the world's population is growing in year \(t\), in billion people per year.

Use the table to estimate \(\int_{0}^{40} f(x) d x\). What values of \(n\) and \(\Delta x\) did you use? $$ \begin{array}{l|c|c|c|c|c} \hline x & 0 & 10 & 20 & 30 & 40 \\ \hline f(x) & 350 & 410 & 435 & 450 & 460 \\ \hline \end{array} $$

Explain in words what the integral represents and give units. \(\int_{0}^{5} s(x) d x\), where \(s(x)\) is rate of change of salinity (salt concentration) in gm/liter per \(\mathrm{cm}\) in sea water, and where \(x\) is depth below the surface of the water in \(\mathrm{cm}\).

The marginal cost function for a company is given by $$ C^{\prime}(q)=q^{2}-16 q+70 \text { dollars/unit, } $$ where \(q\) is the quantity produced. If \(C(0)=500\), find the total cost of producing 20 units. What is the fixed cost and what is the total variable cost for this quantity?

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.

The marginal cost function of producing \(q\) mountain bikes is $$ C^{\prime}(q)=\frac{600}{0.3 q+5} $$ (a) If the fixed cost in producing the bicycles is \(\$ 2000\), find the total cost to produce 30 bicycles. (b) If the bikes are sold for \(\$ 200\) each, what is the profit (or loss) on the first 30 bicycles? (c) Find the marginal profit on the \(31^{\text {st }}\) bicycle.

The following table gives world oil consumption, in billions of barrels per year. ' Estimate total oil consumption during this 25 -year period. $$ \begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 1980 & 1985 & 1990 & 1995 & 2000 & 2005 \\ \hline \text { Oil (bn barrels/yr) } & 22.3 & 21.3 & 23.9 & 24.9 & 27.0 & 29.3 \\\ \hline \end{array} $$

After a foreign substance is introduced into the blood, the rate at which antibodies are made is given by \(r(t)=\frac{t}{t^{2}+1}\) thousands of antibodies per minute, where time, \(t\), is in minutes. Assuming there are no antibodies present at time \(t=0\), find the total quantity of antibodies in the blood at the end of 4 minutes.

The marginal revenue function on sales of \(q\) units of a product is \(R^{\prime}(q)=200-12 \sqrt{q}\) dollars per unit. (a) Graph \(R^{\prime}(q)\). (b) Estimate the total revenue if sales are 100 units. (c) What is the marginal revenue at 100 units? Use this value and your answer to part (b) to estimate the total revenue if sales are 101 units.

World annual natural gas \(^{6}\) consumption, \(N\), in millions of metric tons of oil equivalent, is approximated by \(N=\) \(1770+53 t\), where \(t\) is in years since 1990 . (a) How much natural gas was consumed in 1990 ? In \(2010 ?\) (b) Estimate the total amount of natural gas consumed during the 20 -year period from 1990 to 2010 .

The rate of change of the world's population, in millions of people per year, is given in the following table. (a) Use this data to estimate the total change in the world's population between 1950 and \(2000 .\) (b) The world population was 2555 million people in 1950 and 6085 million people in \(2000 .\) Calculate the true value of the total change in the population. How does this compare with your estimate in part (a)? \begin{equation} \begin{array}{c|c|c|c|c|c|c} \hline \text { Year } & 1950 & 1960 & 1970 & 1980 & 1990 & 2000 \\ \hline \text { Rate of change } & 37 & 41 & 78 & 77 & 86 & 79 \\ \hline \end{array} \end{equation}

Solar photovoltaic (PV) cells are the world's fastestgrowing energy source. \({ }^{7}\) Annual solar \(\mathrm{PV}\) production, \(S\), in megawatts, is approximated by \(S=277 e^{0.368 t}\), where \(t\) is in years since 2000 . Estimate the total solar PV production between 2000 and 2010 .

Ice is forming on a pond at a rate given by $$ \frac{d y}{d t}=\frac{\sqrt{t}}{2} \text { inches per hour, } $$ where \(y\) is the thickness of the ice in inches at time \(t\) measured in hours since the ice started forming. (a) Estimate the thickness of the ice after 8 hours. (b) At what rate is the thickness of the ice increasing after 8 hours?

A village wishes to measure the quantity of water that is piped to a factory during a typical morning. A gauge on the water line gives the flow rate (in cubic meters per hour) at any instant. The flow rate is about \(100 \mathrm{~m}^{3} / \mathrm{hr}\) at \(6 \mathrm{am}\) and increases steadily to about \(280 \mathrm{~m}^{3} / \mathrm{hr}\) at \(9 \mathrm{am} .\) Using only this information, give your best estimate of the total volume of water used by the factory between 6 \(\mathrm{am}\) and \(9 \mathrm{am}\).

(a) Use a graph of the integrand to make a rough estimate of the integral. Explain your reasoning. (b) Use a computer or calculator to find the value of the definite integral. $$ \int_{0}^{1} x^{3} d x $$

The velocity of a car (in miles per hour) is given by \(v(t)=40 t-10 t^{2}\), 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 net worth, \(f(t)\), of a company is growing at a rate of \(f^{\prime}(t)=2000-12 t^{2}\) dollars per year, where \(t\) is in years since \(2005 .\) How is the net worth of the company expected to change between 2005 and 2015 ? If the company is worth \(\$ 40,000\) in 2005, what is it worth in 2015 ?

(a) Use a graph of the integrand to make a rough estimate of the integral. Explain your reasoning. (b) Use a computer or calculator to find the value of the definite integral. $$ \int_{0}^{3} \sqrt{x} d x $$

Your velocity is \(v(t)=\ln \left(t^{2}+1\right) \mathrm{ft} / \mathrm{sec}\) for \(t\) in \(\mathrm{sec}-\) onds, \(0 \leq t \leq 3\). Estimate the distance traveled during this time.

A car initially going 50 ft/sec brakes at a constant rate (constant negative acceleration), coming to a stop in 5 seconds. (a) Graph the velocity from \(t=0\) to \(t=5\). (b) How far does the car travel? (c) How far does the car travel if its initial velocity is doubled, but it brakes at the same constant rate?

(a) Use a graph of the integrand to make a rough estimate of the integral. Explain your reasoning. (b) Use a computer or calculator to find the value of the definite integral. $$ \int_{0}^{1} 3^{t} d t $$

The rate of change of a quantity is given by \(f(t)=t^{2}+1\). Make an underestimate and an overestimate of the total change in the quantity between \(t=0\) and \(t=8\) using (a) \(\quad \Delta t=4\) (b) \(\Delta t=2\) (c) \(\Delta t=1\) What is \(n\) in each case? Graph \(f(t)\) and shade rectangles to represent each of your six answers.

(a) Use a calculator or computer to find \(\int_{0}^{6}\left(x^{2}+1\right) d x\). Represent this value as the area under a curve. (b) Estimate \(\int_{0}^{6}\left(x^{2}+1\right) d x\) using a left-hand sum with \(n=3\). Represent this sum graphically on a sketch of \(f(x)=x^{2}+1\). Is this sum an overestimate or underestimate of the true value found in part (a)? (c) Estimate \(\int_{0}^{6}\left(x^{2}+1\right) d x\) using a right-hand sum with \(n=3\). Represent this sum on your sketch. Is this sum an overestimate or underestimate?

Your velocity is given by \(v(t)=t^{2}+1\) in \(\mathrm{m} / \mathrm{sec}\), with \(t\) in seconds. Estimate the distance, \(s\), traveled between \(t=0\) and \(t=5 .\) Explain how you arrived at your estimate.

A forest fire covers 2000 acres at time \(t=0 .\) The fire is growing at a rate of \(8 \sqrt{t}\) acres per hour, where \(t\) is in hours. How many acres are covered 24 hours later?

The value of a mutual fund increases at a rate of \(R=\) \(500 e^{0.04 t}\) dollars per year, with \(t\) in years since \(2010 .\) (a) Using \(t=0,2,4,6,8,10\), make a table of values for \(R\). (b) Use the table to estimate the total change in the value of the mutual fund between 2010 and 2020 .

Use a calculator or computer to evaluate the integral. $$ \int_{0}^{5} x^{2} d x $$

Water is pumped out of a holding tank at a rate of \(5-5 e^{-0.12 t}\) 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?

(a) Graph \(f(x)=x(x+2)(x-1)\). (b) Find the total area between the graph and the \(x\) -axis between \(x=-2\) and \(x=1\). (c) Find \(\int_{-2}^{1} f(x) d x\) and interpret it in terms of areas.

Use a calculator or computer to evaluate the integral. $$ \int_{1}^{5}(3 x+1)^{2} d x $$

Use a calculator or computer to evaluate the integral. $$ \int_{1}^{4} \frac{1}{\sqrt{1+x^{2}}} d x $$

Use the following table to estimate the area hetween \(f(x)\) and the \(x\) -axis on the interval \(0 \leq x \leq 20\). $$ \begin{array}{r|ccccc} \hline x & 0 & 5 & 10 & 15 & 20 \\ \hline f(x) & 15 & 18 & 20 & 16 & 12 \\ \hline \end{array} $$

Use a calculator or computer to evaluate the integral. $$ \int_{-1}^{1} \frac{1}{e^{t}} d t $$

Compute the definite integral and interpret the result in terms of areas. $$ \int_{1}^{4} \frac{x^{2}-3}{x} d x $$

Use a calculator or computer to evaluate the integral. $$ \int_{1.1}^{1.7} 10(0.85)^{t} d t $$