Exam 6: Integration

A group of marine biologists estimates that if certain conservation measures are implemented, the population of an endangered species of whale will be where N(t) denotes the population at the end of year t. Find the average population of the whales over the next 4 years.

Evaluate the given definite integral.

Let f(x) = 8x Use a Riemann sum with four subintervals of equal length (n = 4) to approximate the area of R (under the graph of f on the interval [0, 2]). Choose the representative points to be the left end points of the subintervals. Repeat previous part with eight subintervals of equal length (n = 8). Compare the approximations obtained in previous parts with the exact area (16 sq units). Do the approximations improve with larger n?

Sketch the graph of the region bounded by the graph of the function f and lines y = 0, x = a and x = b.

Find the indefinite integral. Hint: Let .

Find the indefinite integral.

Find the indefinite integral.

In a study conducted by a certain country's Economic Development Board, it was found that the Lorentz curve for the distribution of income of college teachers was described by the function and that of lawyers by the function Compute the coefficient of inequality for each Lorentz curve. Round your answer to four decimal places. The coefficient of inequality for college teachers is __________. The coefficient of inequality for lawyers is __________. Which profession has a more equitable income distribution?

Find f(x) by solving the initial value problem.

Evaluate the definite integral.

The demand function for a certain brand of compact disc is given by where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Determine the consumers' surplus if the wholesale market price is set at $4/disc. Round your answer to the nearest dollar. $__________

Sketch the graph and find the area of the region bounded by the graph of the function f and lines y = 0, x = a and x = b.

Find the indefinite integral.

Sketch the graph and find the area of the region completely enclosed by the graphs of the given functions f and g.

Suppose an investment is expected to generate income at the rate of dollars/year for the next 5 years. Find the present value of this investment if the prevailing interest rate is 0.05% / year compounded continuously. Round your answer to the nearest dollar. $__________

Find the area of the region under the graph of the function f on the interval [a, b].

The demand function for a certain brand of compact disc is given by The supply function for the compact discs is given by where p is the wholesale unit price in dollars and x is the quantity demanded each week, measured in units of a thousand. Determine the producers' surplus if the wholesale market price is set at the equilibrium price. Round your answer to the nearest dollar. $__________

Evaluate the definite integral.

Sketch the graph and find the area of the region bounded below by the graph of the given function and above by the x-axis from x = a to x = b.

The Armstrongs wish to establish a custodial account to finance their children's education. If they deposit $600 monthly for 8 years in a savings account paying 7% / year compounded continuously, how much will their savings account be worth at the end of this period? Round your answer to the nearest dollar. $__________