Exam 13: A: Definite Integrals - Techniques

The manufacturer of a medicine wants to test how a new 300-milligram capsule is released into the bloodstream. After a volunteer is given a capsule, blood samples are drawn every half-hour, and the number of milligrams of the drug in the bloodstream is calculated. The results obtained are as follows. Use the trapezoidal rule to approximate the average number of milligrams in the bloodstream during the first hours and round your answer to two decimal places. ​

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

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

The manufacturer of a medicine wants to test how a new 300-milligram capsule is released into the bloodstream. After a volunteer is given a capsule, blood samples are drawn every half-hour, and the number of milligrams of the drug in the bloodstream is calculated. The results obtained are as follows. Use the trapezoidal rule to approximate the average number of milligrams in the bloodstream during the first hours and round your answer to two decimal places. ​

Evaluate the definite integral . ​

Evaluate the integral . ​

Find the value of the sum , if . ​

Use an integral formula to evaluate . ​

Evaluate the definite integral . ​

Find the producer's surplus for a product if its demand function is and its supply function is where p is in millions of dollars and x is the number of thousands of units. Round your answer to the nearest million dollars. ​

Find the area of the region enclosed by and . ​​

Evaluate the definite integral . ​

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

The demand function for a product is where p is in millions of dollars and x is the number of thousands of units. If the equilibrium price is $6 million, what is the consumer's surplus? Round your answer to the nearest million dollars. ​

Use the table of values to approximate Use Simpson's Rule and round your answers to one decimal place. ​

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. ​

The following income distribution data define points on Lorenz curves, where x represents the fraction of a certain country's population and L(x) the cumulative percent of income held by fraction x. Use this data and a numerical method to evaluate and hence to find the Gini coefficient of income for this country in 1995 and 2003. Round your answers to three decimal places. ​

Use an integral formula to evaluate . ​

The average life for a particular brand of car battery is given by the following probability density function where t is in years. ​Find the probability that a battery chosen at random lasts between 2 and 7 years. Round your answer to two decimal places. ​

When the area under from to is approximated, the formulas for the sum of n rectangles using left-hand endpoints and right-hand endpoints are: ​ Left-hand endpoints: ​ Right-hand endpoints: ​ Find and . ​