Exam 7: Applications of Integration

Sketch the region bounded by the graphs of the given equations and find the area of that region. y = + 4,y = 2x + 1

Biologists stocked a lake with 700 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake)to be 10800 .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.

Sketch the region enclosed by the curves Find the area of the region correct to two decimal places.

Find the volume of the solid obtained by rotating the region bounded by and about the line

Use the arc length formula to find the length of the curve

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

Find the area of the shaded region.

Find the centroid of the region shown,not by integration,but by locating the centroids of the rectangles and triangles and using additivity of moments.

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

A steady wind blows a kite due west.The kite's height above ground from horizontal position to ft is given by . Find the distance traveled by the kite. Give your answer rounded to two decimal places.

Find the area of the region bounded by the parabola ,the tangent line to this parabola at ,and the x axis.

Calculate the center of mass of the lamina with density

Racing cars driven by Chris and Kelly are side by side at the start of a race.The table shows the velocities of each car (in miles per hour)during the first ten seconds of the race.Use the Midpoint Rule to estimate how much farther Kelly travels than Chris does during the first ten seconds.

Use the method of cylindrical shells to find the volume of solid obtained by rotating the region bounded by the given curves about the x-axis.

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

Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters).You discover that both napkin rings have the same height h as shown in the figure.Use cylindrical shells to compute the volume of a napkin ring created by drilling a hole with radius d through the center of a sphere of radius D and express the answer in terms of t.

Sketch the region enclosed by Find the area of the region.

Find the values of c such that the area of the region bounded by the parabolas is 27.

Use cylindrical shells to find the volume of the solid. A sphere of radius .

Find the volume of the solid obtained by rotating the region bounded by about the x-axis.