Exam 7: Applications of Integration

Find the exact coordinates of the centroid.

Use Simpson's Rule with n = 6 to estimate the length of the curve , .Round your answer to six decimal places.

Find the area of the surface obtained by revolving the graph of y = on [0, 1] about the x-axis.

Find the area of the surface obtained by rotating the curve about the y-axis.

Find the values of c such that the area of the region bounded by the parabolas is .

Solve the initial-value problem.

The Pacific halibut fishery has been modeled by the differential equation where is the biomass (the total mass of the members of the population) in kilograms at time t (measured in years), the carrying capacity is estimated to be and per year. If , find the biomass a year later.

Biologists stocked a lake with fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be . The number of fish tripled in the first year. Assuming that the size of the fish population satisfies the logistic equation, find an expression for the size of the population after t years.

Find the centroid of the region bounded by the given curves.