Exam 6: Applications of the Definite Integral in Geometry, Science and Engineering

Use the method of disks to find the volume of the solid that results by revolving the region enclosed by the curves x = y², x = y + 20, about the y-axis. Approximate area to three decimal places.

Answer true or false. The area of the surface generated by revolving the curve x = t², y = t + 2, 0 $\le$ t $\le$ 1 about the y-axis is given by .

Find the area of the surface generated by revolving , 0 $\le$ y $\le$ 1 about the y-axis. Approximate area to three decimal places.

A spring exerts a force of 1 ton when stretched 5 feet beyond its natural length. How much work is required to stretch the spring 43 feet beyond its natural length?

Use the method of disks to find the volume of the solid that results when the area of the region enclosed by y = x⁵, x = 1, and y = 0 is revolved about the line x = 1.

Evaluate cosh ^{- 1}[cosh(-6)].

Find the centroid of a triangle with vertices (0, 0), (2, 0), and (0, 6).

Find the Mass of a lamina bounded by the graph of y = x² and y = 4. The density of the lamina is = 5.

Find f '(x) if .

Find the arc length of the curve from x = 1 to x = 9.

Find the area of the region enclosed by y = 5 - x² and y = -x + 3 between x = 0 and x = 2.

A 80 foot chain weighing 20 pounds per foot supports a steel beam weighing 2000 pounds. How much work is done in winding 70 feet of the chain onto a drum.

Use a graphing utility to find the region enclosed by the curves x = y³, . Approximate area to three decimal places.

Find if .

Find f '(x) if f(x) = sinh ^{- 1}(6x - 5).

A storage tank is designed by rotating y = -x² + 16, -4 $\le$ x $\le$ 4, about the x-axis where x and y are measured in meters. Use cylindrical shells to determine how many cubic meters the tank will hold. Approximate area to three decimal places.

Find the area of the surface generated when from x = 1 to x = 3 is rotated around the x-axis.

Find the area of the region enclosed by y = 2 -x² and y = -x.

Find if y = sinh(8x + 1).

Use cylindrical shells to find the volume of the solid that results when the area of the region enclosed by y = 2x + 3, x = 2, x = 5, and y = 0 is revolved about the y-axis.