Exam 13: Definite Integrals - Techniques

Approximate the area under the curve over the specified interval by using the indicated number of subintervals (or rectangles) and evaluating the function at the left-hand endpoints of the subintervals. Compute the approximate area using up to 4 decimal places as needed. ​ from to subintervals. ​

A drug manufacturer has developed a time-release capsule with the number of milligrams of the drug in the bloodstream given by where x is in hours and . Find the average number of milligrams of the drug in the bloodstream for the first 3 hours after a capsule is taken. Round your answer to the nearest milligram. ​

Approximate the given integral by using the Trapezoidal Rule with n = 4. Round your answer to 3 decimal places. ​

Evaluate the definite integral . ​

Use Simpson's Rule to approximate with n = 6. Round your answer to two decimal places. ​

Determine the most appropriate method or integral formula for evaluating the given integral. Next, evaluate the integral. ​ I. Integration by parts II. III. IV.

Use Simpson's Rule to approximate with n = 4. Round your answer to 2 decimal places. ​

Evaluate the integral .

Suppose a land developer is planning to dig a small lake and build a group of homes around it. To estimate the cost of the project, the area of the lake must be calculated from the proposed measurements (in feet) given in the following figure and in the table. Use Simpson's Rule to approximate the area of the lake. Round your answer to two decimal places. where a = 300, b = 200, c = 400 ​ ​

The presence of phosphates in certain waste products dumped into a lake promotes the growth of algae. Rampant growth of algae affects the oxygen supply in the water, so an environmental group wishes to estimate the area of algae growth. Group members measure across the algae growth (see the Figure) and obtain the following data (in feet). Use Simpson's Rule to approximate the area of the algae growth and round your answer to the nearest square foot. ​

Find the area of the shaded region for the graph given below.

Evaluate the integral . ​ ​

The demand function for a product is , where p is the number of dollars and x is the number of units. If the equilibrium price is $40, what is the consumer's surplus? ​

For the interval [-5,2] and for n = 3 find h.

The United States' spending for military (in billions of dollars per year) can be modeled by , where represents 1993. Estimate the total spending for the United States military between 2000 and 2005 by using 10 equal subdivisions and right-hand endpoints to approximate the area under the graph of M(t) between and . Round to three decimal places. ​

For a certain product, the total revenue is given by , and the total cost is given by . Write an integral that gives the average profit for the product over the interval from 4 to 11 ​

Suppose that a vending machine company is considering selling some of its machines. Suppose further that the income from these particular machines is a continuous stream with an annual rate of flow at time t given by Find the present value and future value of the machines over the next 3 years if the money is worth 11% compounded continuously. Round answers to the nearest dollar. ​

The demand function for a certain product is given by , where p is the price and q is the number of units demanded. Find the average price as demand ranges from 35 to 81 units. Round your answer to the nearest penny. ​

Assume that the tax burden per capita (in dollars) for selected years from 1950 to 2002 can be modeled by where represents 1950. If the model remains valid, find the predicted average federal tax burden per capita from 2000 to 2010. Round your answer to the nearest dollar. ​

Suppose that the rate at which a nuclear power plant produces radioactive waste is proportional to the number of years it has been operating, according to in pounds per year. Suppose also that the waste decays exponentially at a rate of 9% per year. Then the amount of radioactive waste that will accumulate in b years is given by . Evaluate this integral. ​