Exam 13: A: 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 right-hand endpoints of the subintervals. Compute the approximate area using up to four decimal places as needed. ​ from to subintervals. ​

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

Find the area, if it exists, of the region under the graph of y=f(x) and to the right of x=1. ​ ​

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. ​

Find the area of the shaded region. Round to the nearest hundredth if necessary.

Equations are given whose graphs enclose a region. Find the area of the region. ​ , , and ​

For what value of a is a probability density function? ​

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. ​

Use the function from to and n equal subintervals with the function evaluated at the left-hand endpoints of each subinterval. Find by using the formula for the sum of the areas of the n rectangles (call this S). ​

Find the value of the given sum and round to four decimal places. ​ ​

A 58-year-old couple are considering opening a business of their own. They will either purchase an established Gift and Card Shoppe or open a new Video Rental Palace. The Gift Shoppe has a continuous income stream with an annual rate of flow at time t given by (dollars per year) and the Video Palace has a continuous income stream with a projected annual rate of flow at time t given by (dollars per year). The initial investment is the same for both businesses, and money is worth 10% compounded continuously. Find the present value of the Video Palace over the next 8 years (until the couple reach age 66). Round your answer to the nearest dollar. ​

Find the area, if it exists, of the region under the graph of and to the right of .

Suppose that the demand function for an appliance is where q is the number of units and p is in dollars. What is the consumer's surplus if the equilibrium price is $17.6 and the equilibrium quantity is 21? Round your answer to the nearest dollar. ​

Evaluate the definite integral . ​

Use Simpson's Rule to approximate with n = 6. Round your answer to 3 decimal places. ​

Find the value of the sum ​

Market revenue for Hammer Inc. (in millions of dollars per year) can be modeled by , where represents 1990. Evaluate . Round your answer to the nearest cent. ​

Find the average value of the given function over the given interval. ​

Use integration by parts to evaluate . Note that evaluation may require integration by parts more than once. ​ ​

Find the area of the shaded region. ​ ​