Chapter 5

In the country of Equalia, the Gini index for income is, by law, fixed at \(0 .\) State one good thing and one bad thing about living in Equalia.

Evaluate each definite integral. $$ \int_{0}^{1}\left(x^{99}+x^{9}+1\right) d x $$

Find each integral. [Hint: Try some algebra.] $$ \int(x+4)(x-2) d x $$

A culture of bacteria is growing at the rate of \(20 e^{0.8 t}\) cells per day, where \(t\) is the number of days since the culture was started. Suppose that the culture began with 50 cells. a. Find a formula for the total number of cells in the culture after \(t\) days. b. If the culture is to be stopped when the population reaches 500 , when will this occur?

Find the area bounded by the given curves. \(y=3 x^{2}-x-1\) and \(y=5 x+8\)

Does a higher Gini index mean more equality or more inequality?

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

Find each integral. [Hint: Try some algebra.] $$ \int(x+1)^{2} x^{3} d x $$

An ice cube tray filled with tap water is placed in the freezer, and the temperature of the water is changing at the rate of \(-12 e^{-0.2 t}\) degrees Fahrenheit per hour after \(t\) hours. The original temperature of the tap water was 70 degrees. a. Find a formula for the temperature of water that has been in the freezer for \(t\) hours. b. When will the ice be ready? (Water freezes at 32 degrees. \()\)

Find the area bounded by the given curves. \(y=3 x^{2}-12 x\) and \(y=0\)

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

Find each integral. [Hint: Try some algebra.] $$ \int(x-1)^{2} \sqrt{x} d x $$

The divorce rate in the United States (divorces per year) has been declining in recent years. The number of divorces per year is predicted to be \(0.94 e^{-0.02 t}\) million, where \(t\) is the number of years since 2014 . a. Find a formula for the total number of divorces within \(t\) years of 2014 b. Use your formula to find the total number of divorces from 2014 to \(2020 .\) Source: National Center for Health Statistics

Find the area bounded by the given curves. \(y=x^{2}-6 x\) and \(y=0\)

Evaluate each definite integral. $$ \int_{-2}^{2}\left(3 w^{2}-2 w\right) d w $$

A factory installs new equipment that is expected to generate savings at the rate of \(800 e^{-0.2 t}\) dollars per year, where \(t\) is the number of years that the equipment has been in operation. a. Find a formula for the total savings that the equipment will generate during its first \(t\) years. b. If the equipment originally cost \(\$ 2000\), when will it "pay for itself"?

Find the area bounded by the given curves. \(y=3 x^{2}\) and \(y=12\)

Evaluate each definite integral. $$ \int_{1}^{4} \frac{1}{y^{2}} d y $$

A company installs a new computer that is expected to generate savings at the rate of \(20,000 e^{-0.02 t}\) dollars per year, where \(t\) is the number of years that the computer has been in operation. a. Find a formula for the total savings that the computer will generate during its first \(t\) years. b. If the computer originally cost \(\$ 250,000\), when will it "pay for itself"?

Find the area bounded by the given curves. \(y=x^{2}\) and \(y=4\)

Evaluate each definite integral. $$ \int_{1}^{4} \frac{1}{\sqrt{z}} d z $$

For each definite integral: a. Evaluate it "by hand." b. Check your answer by using a graphing calculator. $$ \int_{0}^{2} e^{x^{3}} x^{2} d x $$

For each definite integral: a. Evaluate it "by hand." b. Check your answer by using a graphing calculator. $$ \int_{0}^{1} \frac{x}{x^{2}+1} d x $$

A real estate investment, originally worth \(\$ 5000\), grows continuously at the rate of \(400 e^{0.05 t}\) dollars per year, where \(t\) is the number of years since the investment was made. a. Find a formula for the value of the investment after \(t\) years. b. Use your formula to find the value of the investment after 10 years.

Find the area bounded by the given curves. \(y=x^{2}\) and \(y=x^{3}\)

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

For each definite integral: a. Evaluate it "by hand." b. Check your answer by using a graphing calculator. $$ \int_{2}^{3} \frac{x^{2}}{x^{3}-7} d x $$

A biotechnology investment, originally worth \(\$ 20,000\), grows continuously at the rate of \(1000 e^{0.10 t}\) dollars per year, where \(t\) is the number of years since the investment was made. a. Find a formula for the value of the investment after years. b. Use your formula to find the value of the investment after 7 years.

Find the area bounded by the given curves. \(y=x^{3}\) and \(y=x^{4}\)

Evaluate each definite integral. $$ \int_{1}^{e^{2}} \frac{3}{x} d x $$

For each definite integral: a. Evaluate it "by hand." b. Check your answer by using a graphing calculator. $$ \int_{0}^{4} \sqrt{x^{2}+9} x d x $$

World consumption of lead is running at the rate of \(5 e^{0.05 t}\) million metric tons per year, where \(t\) is measured in years and \(t=0\) corresponds to 2014 a. Find a formula for the total amount of lead that will be consumed within \(t\) years of 2014 b. Use a graphing calculator to find when the world's known resources of 89 million metric tons of lead will be exhausted. [Hint: Use INTERSECT.] Lead has many uses, from batteries to shields against radioactivity.

Find the area bounded by the given curves. \(y=4 x^{3}+3\) and \(y=4 x+3\)

Evaluate each definite integral. $$ \int_{1}^{3}\left(9 x^{2}+x^{-1}\right) d x $$

For each definite integral: a. Evaluate it "by hand." b. Check your answer by using a graphing calculator. $$ \int_{0}^{3} \sqrt{x^{2}+16} x d x $$

Find the area bounded by the given curves. \(y=x^{3}\) and \(y=4 x\)

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

A homeowner installs a solar water heater that is expected to generate savings at the rate of \(70 e^{0.03 t}\) dollars per year, where \(t\) is the number of years since it was installed. a. Find a formula for the total savings within the first \(t\) years of operation. b. Use a graphing calculator to find when the heater will "pay for itself" if it cost \$800. [Hint: Use INTERSECT.]

For each definite integral: a. Evaluate it "by hand." b. Check your answer by using a graphing calculator. $$ \int_{2}^{3} \frac{d x}{1-x} $$

Subscriptions to satellite radio have been growing rapidly since they are now included in many new cars. The rate of increase has been approximately \(2.15 e^{0.086 t}\) million subscriptions per year, where \(t\) is the number of years since 2014 . a. Use this rate to find a formula to predict the total increase in subscriptions within \(t\) years of 2014 b. Use your formula to find the total increase from 2014 to 2020

Find the area bounded by the given curves. \(y=7 x^{3}-36 x\) and \(y=3 x^{3}+64 x\)

Evaluate each definite integral. $$ \int_{-2}^{-1} 3 x^{-1} d x $$

For each definite integral: a. Evaluate it "by hand." b. Check your answer by using a graphing calculator. $$ \int_{3}^{4} \frac{d x}{2-x} $$

The value of a recently issued General Electric bond increases in value at the rate of \(40 e^{0.04 t}\) dollars per year, where \(t=0\) represents \(2013 .\) a. Find a formula for the total increase in the value of the stock within \(t\) years of \(2013 .\) b. Use your formula to find the total increase from 2013 to 2028 .

Find the area bounded by the given curves. \(y=x^{n}\) and \(y=x^{n-1}(\) for \(n>1)\)

Evaluate each definite integral. $$ \int_{-3}^{-1}\left(1+x^{-1}\right) d x $$

For each definite integral: a. Evaluate it "by hand." b. Check your answer by using a graphing calculator. $$ \int_{1}^{8} \frac{e^{\sqrt[3]{x}}}{\sqrt[3]{x^{2}}} d x $$

\(59-63 .\) Choose the correct answer. \(\int e^{x} d x=?\) a. \(\frac{1}{x+1} e^{x+1}+C\) b. \(e^{x}+C\) c. \(e^{x} x+C\)

Find the area bounded by the given curves. \(y=e^{x}\) and \(y=x+3\)

Evaluate each definite integral. $$ \int_{0}^{1} 12 e^{3 x} d x $$