Chapter 11

Find the average value of the function f over the indicated interval \([a, b]\). $$f(x)=4-x^{2} ;[-2,3]$$

Evaluate the definite integral. $$\int_{0}^{1} \sqrt{2 x}(\sqrt{x}+\sqrt{2}) d x$$

Find the indefinite integral. $$\int \frac{1}{x(\ln x)^{2}} d x$$

Find the indefinite integral. $$\int \frac{x^{4}-1}{x^{2}} d x$$

Sketch the graph and find the area of the region completely enclosed by the graphs of the given functions \(f\) and \(g\). $$f(x)=x^{3}-6 x^{2}+9 x\( and \)g(x)=x^{2}-3 x$$

Find the average value of the function f over the indicated interval \([a, b]\). $$f(x)=x^{2}+2 x-3 ;[-1,2]$$

Evaluate the definite integral. $$\int_{1}^{4} \frac{3 x^{3}-2 x^{2}+4}{x^{2}} d x$$

Find the indefinite integral. $$\int \frac{\sqrt{\ln x}}{x} d x$$

Find the indefinite integral. $$\int(2 t+1)(t-2) d t$$

Find the average value of the function f over the indicated interval \([a, b]\). $$f(x)=x^{3} ;[-1,1]$$

Evaluate the definite integral. $$\int_{1}^{2}\left(1+\frac{1}{u}+\frac{1}{u^{2}}\right) d u$$

Find the indefinite integral. $$\int \frac{(\ln x)^{7 / 2}}{x} d x$$

Find the indefinite integral. $$\int u^{-2}\left(1-u^{2}+u^{4}\right) d u$$

Sketch the graph and find the area of the region completely enclosed by the graphs of the given functions \(f\) and \(g\). $$f(x)=x \sqrt{9-x^{2}}\( and \)g(x)=0$$

Find the average value of the function f over the indicated interval \([a, b]\). $$f(x)=\sqrt{2 x+1} ;[0,4]$$

A division of Ditton Industries manufactures a deluxe toaster oven. Management has determined that the daily marginal cost function associated with producing these toaster ovens is given by $$ C^{\prime}(x)=0.0003 x^{2}-0.12 x+20 $$ where \(C^{\prime}(x)\) is measured in dollars/unit and \(x\) denotes the number of units produced. Management has also determined that the daily fixed cost incurred in the production is \(\$ 800 .\) a. Find the total cost incurred by Ditton in producing the first 300 units of these toaster ovens per day. b. What is the total cost incurred by Ditton in producing the 201st through 300th units/day?

Find the indefinite integral. $$\int\left(x e^{x^{2}}-\frac{x}{x^{2}+2}\right) d x$$

Find the indefinite integral. $$\int \frac{1}{x^{2}}\left(x^{4}-2 x^{2}+1\right) d x$$

Sketch the graph and find the area of the region completely enclosed by the graphs of the given functions \(f\) and \(g\). $$f(x)=2 x\( and \)g(x)=x \sqrt{x+1}$$

Find the average value of the function f over the indicated interval \([a, b]\). $$f(x)=e^{-x} ;[0,4]$$

The management of Ditton Industries has determined that the daily marginal revenue function associated with selling \(x\) units of their deluxe toaster ovens is given by $$ R^{\prime}(x)=-0.1 x+40 $$ where \(R^{\prime}(x)\) is measured in dollars/unit. a. Find the daily total revenue realized from the sale of 200 units of the toaster oven. b. Find the additional revenue realized when the production (and sales) level is increased from 200 to 300 units.

Find the indefinite integral. $$\int\left(x e^{-x^{2}}+\frac{e^{x}}{e^{x}+3}\right) d x$$

Find the indefinite integral. $$\int \sqrt{t}\left(t^{2}+t-1\right) d t$$

Find the average value of the function f over the indicated interval \([a, b]\). $$f(x)=x e^{x^{2}} ;[0,2]$$

Refer to Exercise \(41 .\) The daily marginal profit function associated with the production and sales of the deluxe toaster ovens is known to be $$ P^{\prime}(x)=-0.0003 x^{2}+0.02 x+20 $$ where \(x\) denotes the number of units manufactured and sold daily and \(P^{\prime}(x)\) is measured in dollars/unit. a. Find the total profit realizable from the manufacture and sale of 200 units of the toaster ovens per day. Hint: \(P(200)-P(0)=\int_{0}^{200} P^{\prime}(x) d x, P(0)=-800\). b. What is the additional daily profit realizable if the production and sale of the toaster ovens are increased from 200 to 220 units/day?

Find the indefinite integral. $$\int \frac{x+1}{\sqrt{x}-1} d x$$

Find the indefinite integral. $$\int \frac{d s}{(s+1)^{-2}}$$

Find the average value of the function f over the indicated interval \([a, b]\). $$f(x)=\frac{1}{x+1} ;[0,2]$$

U.S. Internet advertising revenue grew at the rate of $$ R(t)=0.82 t+1.14 \quad(0 \leq t \leq 4) $$ billion dollars/year between \(2002(t=0)\) and \(2006(t=4)\). The advertising revenue in 2002 was \(\$ 5.9\) billion. a. Find an expression giving the advertising revenue in year \(t\). b. If the trend continued, what was the Internet advertising revenue in \(2007 ?\)

Find the indefinite integral. $$\int \frac{e^{-u}-1}{e^{-u}+u} d u$$

Find the indefinite integral. \$$\int\left(\sqrt{x}+\frac{3}{x}-2 e^{x}\right) d x$$

A study proposed in 1980 by researchers from the major producers and consumers of the world's coal concluded that coal could and must play an important role in fueling global economic growth over the next \(20 \mathrm{yr}\). The world production of coal in 1980 was \(3.5\) billion metric tons. If output increased at the rate of \(3.5 e^{0.08 t}\) billion metric tons/year in year \(t(t=0\) corresponding to 1980 ), determine how much coal was produced worldwide between 1980 and the end of the 20 th century.

Mobile-phone ad spending is expected to grow at the rate of $$ R(t)=0.8256 t^{-0.04} \quad(1 \leq t \leq 5) $$ billion dollars/year between \(2007(t=1)\) and \(2011(t=5)\). The mobile-phone ad spending in 2007 was \(\$ 0.9\) billion. a. Find an expression giving the mobile-phone ad spending in year \(t\). b. If the trend continued, what will be the mobile-phone ad spending in 2012 ?

Find the indefinite integral. $$\int x(x-1)^{5} d x$$

Find the indefinite integral. $$\int\left(e^{t}+t^{e}\right) d t$$

A bottle of white wine at room temperature \(\left(68^{\circ} \mathrm{F}\right)\) is placed in a refrigerator at 4 p.m. Its temperature after \(t\) hr is changing at the rate of $$ -18 e^{-0.6 r} $$ \({ }^{\circ} \mathrm{F}\) /hour. By how many degrees will the temperature of the wine have dropped by \(7 \mathrm{p} . \mathrm{m} . ?\) What will the temperature of the wine be at \(7 \mathrm{p} \cdot \mathrm{m}\).?

Tempco Electronics, a division of Tempco Toys, manufactures an electronic football game. An efficiency study showed that the rate at which the games are assembled by the average worker \(t\) hr after starting work at 8 a.m. is $$ -\frac{3}{2} t^{2}+6 t+20 \quad(0 \leq t \leq 4) $$ units/hour. a. Find the total number of games the average worker can be expected to assemble in the 4 -hr morning shift. b. How many units can the average worker be expected to assemble in the first hour of the morning shift? In the second hour of the morning shift?

Find the indefinite integral. $$\int \frac{t}{t+1} d t$$

Find the indefinite integral. $$\int\left(\frac{1}{x^{2}}-\frac{1}{\sqrt[3]{x^{2}}}+\frac{1}{\sqrt{x}}\right) d x$$

FLow The net investment flow (rate of capital formation) of the giant conglomerate LTF incorporated is projected to be $$ t \sqrt{\frac{1}{2} t^{2}+1} $$ million dollars/year in year \(t\). Find the accruement on the companv's capital stock in the second year.

\(\ln\) a recent pretrial run for the world water speed record, the velocity of the Sea Falcon II \(t\) sec after firing the booster rocket was given by $$ v(t)=-t^{2}+20 t+440 \quad(0 \leq t \leq 20) $$ feet/second. Find the distance covered by the boat over the 20 -sec period after the booster rocket was activated. Hint: The distance is given by \(\int_{0}^{20} v(t) d t\).

Find the indefinite integral. $$\int \frac{1-\sqrt{x}}{1+\sqrt{x}} d x$$

Find the indefinite integral. $$\int\left(\frac{x^{3}+x^{2}-x+1}{x^{2}}\right) d x$$

Based on a preliminary report by a geological survey team, it is estimated that a newly discovered oil field can be expected to produce oil at the rate of $$ R(t)=\frac{600 t^{2}}{t^{3}+32}+5 \quad(0 \leq t \leq 20) $$ thousand barrels/year, \(t\) yr after production begins. Find the amount of oil that the field can be expected to yield during the first 5 yr of production, assuming that the projection holds true.

Annual sales (in millions of units) of pocket computers are expected to grow in accordance with the function $$ f(t)=0.18 t^{2}+0.16 t+2.64 \quad(0 \leq t \leq 6) $$ where \(t\) is measured in years, with \(t=0\) corresponding to \(1997 .\) How many pocket computers were sold over the 6 -yr period between the beginning of 1997 and the end of \(2002 ?\)

Find the indefinite integral. $$\int \frac{1+\sqrt{x}}{1-\sqrt{x}} d x$$

Find the indefinite integral. $$\int \frac{t^{3}+\sqrt[3]{t}}{t^{2}} d t$$

In tests conducted by Auto Test Magazine on two identical models of the Phoenix Elite - one equipped with a standard engine and the other with a turbo-charger-it was found that the acceleration of the former is given by $$ a=f(t)=4+0.8 t \quad(0 \leq t \leq 12) $$ \(\mathrm{ft} / \mathrm{sec} / \mathrm{sec}, t\) sec after starting from rest at full throttle, whereas the acceleration of the latter is given by $$ a=g(t)=4+1.2 t+0.03 t^{2} \quad(0 \leq t \leq 12) $$ ft/sec/sec. How much faster is the turbo-charged model moving than the model with the standard engine at the end of a 10 -sec test run at full throttle?

Suppose a tractor purchased at a price of $$\$ 60,000$$ is to be depreciated by the double declining-balance method over a 10 -yr period. It can be shown that the rate at which the book value will be decreasing is given by $$ R(t)=13388.61 e^{-0.22314} \quad(0 \leq t \leq 10) $$ dollars/year at year \(t\). Find the amount by which the book value of the tractor will depreciate over the first \(5 \mathrm{yr}\) of its life.

The percentage of families with children that are headed by single females grew at the rate of $$ R(t)=0.8499 t^{2}-3.872 t+5 \quad(0 \leq t \leq 3) $$ households/decade between \(1970(t=0)\) and \(2000(t=3)\). The number of such households stood at \(5.6 \%\) of all families in 1970 . a. Find an expression giving the percentage of these households in the \(t\) h decade. b. If the trend continued, estimate the percentage of these households in 2010 . c. What was the net increase in the percentage of these households from 1970 to \(2000 ?\)