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GENERAL: Consumption of Natural 91Ó°ÊÓ World consumption of lead is running at the rate of \(4.1 e^{0.05 t}\) million metric tons per year, where \(t\) is measured in years and \(t=0\) corresponds to 2010 . a. Find a formula for the total amount of lead that will be consumed within \(t\) years of \(2010 .\) b. Use a graphing calculator to find when the world's known resources of 80 million metric tons of lead will be exhausted. [Hint: Use INTERSECT.] Lead has many uses, from batteries to shields against radioactivity.

BUSINESS: Income In 2010 Apple's net income was 6 billion dollars and growing at the rate of \(0.2 x+1.6\) billion dollars per year, where \(x\) is the number of years since 2010 . Find a formula to predict Apple's net income at any time \(x\) and use your formula to predict their income in \(2020 .\)

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. $$ \int_{0}^{4} \sqrt{x^{2}+9} x d x $$

GENERAL: Total Savings 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 \(\$ 800\).

BUSINESS: Revenue In 2010 , revenues at Research In Motion (maker of the BlackBerry communications device) were \(5.5\) billion dollars and growing at the rate of \(0.2 x+1.5\) billion dollars per year, where \(x\) stands for the number of years since 2010 . Find a formula for Research In Motion's revenues at any time \(x\) and use your formula to predict their revenues in \(2020 .\)

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

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

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

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