Exam 6: Integration

A drug is carried into an organ of volume by a liquid that enters the organ at the rate of and leaves it at the rate of .The concentration of the drug in the liquid entering the organ is .If the concentration of the drug in the organ at time is increasing at the rate of , and the concentration of the drug in the organ initially is , show that the concentration of the drug in the organ at time is given by

The population of a certain city is projected to grow at the rate of people/year, t years from now.The current population is 70,000.What will be the population 5 years from now? __________ people

Find the indefinite integral.

Figure (a) shows a vacant lot with a 80-ft frontage in a development.To estimate its area, we introduce a coordinate system so that the x-axis coincides with the edge of the straight road forming the lower boundary of the property, as shown in Figure (b).Then, thinking of the upper boundary of the property as the graph of a continuous function f over the interval [0, 80], we see that the problem is mathematically equivalent to that of finding the area under the graph of f on [0, 80].To estimate the area of the lot using a Riemann sum, we divide the interval [0, 80] into four equal subintervals of length 20 ft.Then, using surveyor's equipment, we measure the distance from the midpoint of each of these subintervals to the upper boundary of the property.These measurements give the values of f(x) at x = 10, 30, 50, and 70.What is the approximate area of the lot? __________ square feet

Find the indefinite integral.

U.S.Internet advertising revenue grew at the rate of billion dollars/year between 2002 (t = 0) and 2006 (t = 4).The advertising revenue in 2002 was $6.3 billion.Find an expression giving the advertising revenue in year t.If the trend continued, what was the Internet advertising revenue in 2007?

The number of Americans aged 45 to 54 (which stood at 25 million at the beginning of 1990) grew at the rate of million people/year, t years from the beginning of 1990.How many Americans aged 45 to 54 were added to the population between 1990 and the year 1995? Give your answer to one decimal place.

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 ft/sec, t seconds after starting from rest at full throttle, whereas the acceleration of the latter is given by ft/sec.How much faster is the turbo-charged model moving than the model with the standard engine at the end of a 10-second test run at full throttle? __________ ft/sec

Find the area of the region under the graph of the function on the interval .

Evaluate the given definite integral.

Find the indefinite integral.

It is known that the quantity demanded of a certain make of portable hair dryer is x hundred units/week and the corresponding wholesale unit price is dollars.Determine the consumers' surplus if the wholesale market price is set at $20/unit.Round your answer to the nearest dollar.

Evaluate the given definite integral.

Find the area of the region under the graph of the function on the interval , using the fundamental theorem of calculus.Then verify your result using geometry.

Annual sales (in millions of units) of hand-held computers are expected to grow in accordance with the function where t is measured in years, with t = 0 corresponding to 1997.How many hand-held computers will be sold over the 6-year period between the beginning of 1997 and the end of 2002?

Find the indefinite integral.

Suppose an investment is expected to generate income at the rate of dollars/year for the next 6 years.Find the present value of this investment if the prevailing interest rate is 8% / year compounded continuously.Round your answer to the nearest dollar.

Find the indefinite integral.

Find the indefinite integral.

The demand function for a certain make of replacement cartridges for a water purifier is given by where p is the unit price in dollars and x is the quantity demanded each week, measured in units of a thousand.Determine the consumers' surplus if the market price is set at $5/cartridge.Round your answer to the nearest dollar.