Chapter 5

For each definite integral: a. Evaluate it "by hand." b. Check your answer by using a graphing calculator. $$ \int_{1}^{4} \frac{e^{\sqrt{x}}}{\sqrt{x}} d x $$

Choose the correct answer. \(\int x^{-1} d x=?\) a. \(\ln |x|+C\) b. \(\frac{1}{0} x^{0}+C\) c. \(x^{-1} x+C\)

Find the area bounded by the given curves. \(y=\ln x\) and \(y=x-2\)

Evaluate each definite integral. $$ \int_{0}^{2} 3 e^{x / 2} d x $$

In 2000 the birthrate in Africa increased from \(17 e^{0.02 t}\) million births per year to \(22 e^{0.02 t}\) million births per year, where \(t\) is the number of years since 2000 . Find the total increase in population that will result from this higher birthrate between \(2000(t=0)\) and \(2050(t=50)\).

Prove the integration formula $$ \int u^{n} d u=\frac{1}{n+1} u^{n+1}+C \quad(n \neq-1) $$ as follows. a. Differentiate the right-hand side of the formula with respect to \(x\) (remembering that \(u\) is a function of \(x\) ). b. Verify that the result of part (a) agrees with the integrand in the formula (after replacing \(d u\) in the formula by \(u^{\prime} d x\) ).

Choose the correct answer. \(\int x^{e} d x=?\) a. \(\frac{1}{e+1} x^{e+1}+C\) b. \(x^{e}+C\) c. \(x^{e} x+C\)

Evaluate each definite integral. $$ \int_{-1}^{1} 5 e^{-x} d x $$

A company expects profits of \(60 e^{0.02 t}\) thousand dollars per month, but predicts that if it builds a new and larger factory, its profits will be \(80 e^{0.04 t}\) thousand dollars per month, where \(t\) is the number of months from now. Find the extra profits resulting from the new factory during the first two years \((t=0\) to \(t=24\) ). If the new factory will cost \(\$ 1,000,000\), will this cost be paid off during the first two years?

Choose the correct answer. \(\int e^{3} d x=?\) a. \(\frac{1}{4} e^{4}+C\) b. \(e^{3}+C\) c. \(e^{3} x+C\)

Evaluate each definite integral. $$ \int_{0}^{1}\left(6 x^{2}-4 e^{2 x}\right) d x $$

A factory installs new machinery that saves \(S(x)=1200-20 x\) dollars per year, where \(x\) is the number of years since installation. However, the cost of maintaining the new machinery is \(C(x)=100 x\) dollars per year. a. Find the year \(x\) at which the maintenance \(\operatorname{cost} C(x)\) will equal the savings \(S(x)\). (At this time, the new machinery should be replaced.) b. Find the accumulated net savings [savings \(S(x)\) minus \(\operatorname{cost} C(x)]\) during the period from \(t=0\) to the replacement time found in part (a).

Choose the correct answer. \(\int e^{-1} d x=?\) a. \(\ln |e|+C\) b. \(\frac{1}{0} e^{0}+C\) c. \(e^{-1} x+C\)

Evaluate each definite integral. $$ \int_{\ln 2}^{\ln 3} e^{x} d x $$

Find \(\int(x+1) d x\) : a. By using the formula for \(\int u^{n} d u\) with \(n=1\). b. By dropping the parentheses and integrating directly. c. Can you reconcile the two seemingly different answers? [Hint: Think of the arbitrary constant.]

Evaluate each definite integral. $$ \int_{0}^{\ln 5} e^{x} d x $$

A country's imports are \(I(t)=30 e^{0.2 t}\) and its exports are \(E(t)=25 e^{0.1 t},\) both in billions of dollars per year, where \(t\) is measured in years and \(t=0\) corresponds to the beginning of 2000\. Find the country's accumulated trade deficit (imports minus exports) for the 10 years beginning with 2000.

A company's marginal cost function is \(M C(x)=\frac{1}{2 x+1}\) and its fixed costs are \(50 .\) Find the cost function.

Find a formula for \(\int e^{a x+b} d x\) where \(a\) and \(b\) are constants.

Evaluate each definite integral. $$ \int_{1}^{2} \frac{(x+1)^{2}}{x} d x $$

An employer offers to pay workers at the rate of \(30,000 e^{0.04 t}\) dollars per year, while the union demands payment at the rate of \(30,000 e^{0.08 t}\) dollars per year, where \(t=0\) corresponds to the beginning of the contract. Find the accumulated difference in pay between these two rates over the 10 -year life of the contract.

A company's marginal cost function is \(M C(x)=\frac{1}{\sqrt{2 x+25}}\) and its fixed costs are \(100 .\) Find the cost function.

Evaluate each definite integral. $$ \int_{1}^{2} \frac{(x+1)^{2}}{x^{2}} d x $$

The population of a city is expected to be \(P(x)=x\left(x^{2}+36\right)^{-1 / 2}\) million people after \(x\) years. Find the average population between year \(x=0\) and year \(x=8\).

Use a graphing calculator to evaluate each definite integral, rounding answers to three decimal places. [Hint: Use a command like FnInt or \(\left.\int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x} .\right]\) $$ \int_{0}^{2} \frac{1}{x^{2}+1} d x $$

A company's marginal revenue function is \(M R(x)=700 x^{-1}\) and its marginal cost function is \(M C(x)=500 x^{-1}\) (both in thousands of dollars), where \(x\) is the number of units \((x>1)\). Find the total profit from \(x=100\) to \(x=200\)

A company's marginal revenue function is \(M R(x)=700 x^{-1}\) and its marginal cost function is \(M C(x)=500 x^{-1}\) (both in thousands of dollars), where \(x\) is the number of units \((x>1)\). Find the total profit from \(x=200\) to \(x=300\)

Find the area between the curve \(y=x e^{x^{2}}\) and the \(x\) -axis from \(x=1\) to \(x=3\) (Leave the answer in its exact form.)

Which one of these formulas is correct? a. \(\int \ln x d x=\frac{1}{|x|}+C\) b. \(\int \ln |x| d x=\frac{1}{x}+C\) c. \(\int \frac{1}{x} d x=\ln |x|+C\) d. \(\int \frac{1}{\ln x} d x=|x|+C\)

Use a graphing calculator to evaluate each definite integral, rounding answers to three decimal places. [Hint: Use a command like FnInt or \(\left.\int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x} .\right]\) $$ \int_{-1}^{1} \sqrt{x^{4}+1} d x $$

Seat Belts Seat belt use in the United States has risen to \(86 \%,\) but nonusers still risk needless expense and serious injury. The upper curve in the following graph represent an estimate of fatalities per year if seat belts were not used, and the lower curve is a prediction of actual fatalities per year with seat belt use, both in thousands ( \(x\) represents years after 2010). Therefore, the area between the curves represents lives saved by seat belts. Find the area between the curves from 0 to 20 , giving an estimate of the number of lives saved by seat belts during the years \(2010-2030\).

A company's sales (in millions) during week \(x\) are given by \(S(x)=\frac{1}{x+1} .\) Find the average sales from week \(x=1\) to week \(x=4\).

Use a graphing calculator to evaluate each definite integral, rounding answers to three decimal places. [Hint: Use a command like FnInt or \(\left.\int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x} .\right]\) $$ \int_{-1}^{1} e^{x^{2}} d x $$

Find the derivative of each function. \(e^{x^{3}+6 x}\)

A subject can perform a task at the rate of \(\sqrt{2 t+1}\) tasks per minute at time \(t\) minutes. Find the total number of tasks performed from time \(t=0\) to time \(t=12\).

Use a graphing calculator to evaluate each definite integral, rounding answers to three decimal places. [Hint: Use a command like FnInt or \(\left.\int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x} .\right]\) $$ \int_{-2}^{2} e^{(-1 / 2) x^{2}} d x $$

Find the derivative of each function. \(e^{x^{2}+5 x}\)

An experimental drug lowers a patient's blood serum cholesterol at the rate of \(t \sqrt{25}-t^{2}\) units per day, where \(t\) is the number of days since the drug was administered \((0 \leq t \leq 5)\). Find the total change during the first 3 days.

Use a graphing calculator to evaluate each definite integral, rounding answers to three decimal places. [Hint: Use a command like FnInt or \(\left.\int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x} .\right]\) $$ \int_{0}^{4} \sqrt{x} e^{x} d x $$

Find the derivative of each function. \(\ln \left(x^{3}+6 x\right)\)

BUSINESS: Total Sales During an automobile sale, cars are selling at the rate of \(\frac{12}{x+1}\) cars per day, where \(x\) is the number of days since the sale began. How many cars will be sold during the first 7 days of the sale?

Use a graphing calculator to evaluate each definite integral, rounding answers to three decimal places. [Hint: Use a command like FnInt or \(\left.\int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x} .\right]\) $$ \int_{1}^{4} x^{x} d x $$

Find the derivative of each function. $\ln \left(x^{2}+5 x\right)

A real estate office is selling condominiums at the rate of \(100 e^{-x / 4}\) per week after \(x\) weeks. How many condominiums will be sold during the first 8 weeks?

a. Evaluate the definite integral \(\int_{0}^{3} x^{2} d x\). b. Evaluate the same definite integral by completing the following calculation, in which the antiderivative includes a constant \(C\). $$ \int_{0}^{3} x^{2} d x=\left.\left(\frac{1}{3} x^{3}+C\right)\right|_{0} ^{3}=\cdots $$ [The constant \(C\) should cancel out, giving the same answer as in part (a).] c. Explain why the constant will cancel out of any definite integral. (We therefore omit the constant in definite integrals. However, be sure to keep the \(+C\) in indefinite integrals.)

Find the derivative of each function. \(\sqrt{x^{3}+1}\)

An aircraft company estimates its marginal revenue function for helicopters to be \(M R(x)=(x+40) \sqrt{x^{2}+80 x}\) thousand dollars, where \(x\) is the number of helicopters sold. Find the total revenue from the sale of the first 10 helicopters.

Show that for any number \(a>0\) $$ \int_{1}^{a} \frac{1}{x} d x=\ln a $$

\(75-76 .\)A factory is discharging pollution into a lake at the rate of \(r(t)\) tons per year given below, where \(t\) is the number of years that the factory has been in operation. Find the total amount of pollution discharged during the first 3 years of operation. $$ r(t)=\frac{t}{t^{2}+1} $$

A factory is discharging pollution into a lake at the rate of \(r(t)\) tons per year given below, where \(t\) is the number of years that the factory has been in operation. Find the total amount of pollution discharged during the first 3 years of operation. $$ r(t)=t \sqrt{t^{2}+16} $$