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

A dredger scoops a shovel full of mud weighing 4000 pounds from the bottom of a river at a constant rate. Water leaks out uniformly at such a rate that half the weight of the contents is lost when the scoop has been lifted 25 feet. How much work is done by the dredger in lifting the mud this distance?

Evaluate .

Answer true or false. = 1

Find the arc length of the curve from x = 1 to x = 4.

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

Find the Mass of a lamina bounded by the y-axis and the graph of x = 81 - y⁴. The density of the lamina is = 3.

Approximate the area of the surface generated by revolving the curve y = 3 cos x, , about the x-axis. Approximate area to three decimal places.

A swimming pool is 24 feet long and 16 feet wide. The bottom is flat but inclined as shown in the figure. The water is 12 feet deep on one end and 2 feet deep on the other. Find the force on the bottom of the pool when it is filled with water (weight density 62.4 lbs/ft³).

Find the arc length of the curve 4y³ = 9x² from (0, 0) to .

Use the method of washers to find the volume of the solid that results when the area of the region enclosed by , y = 0, and x = 16 is revolved about the y axis.

Find the area of the region enclosed by the curves y = 2x² and y = 2x by integrating with respect to x. Approximate area to three decimal places.

A force of 7x lb is applied to an object as it moves from x = 3 ft to x = 4 ft. How much work is done?

Find the work done when a variable force of N in the positive x-direction moves an object from x = 4 m to x = 10 m.

Find f '(x) if f(x) = (tanh ^{- 1}2x)².

Find the area between the curves y = |x - 4|, . Approximate area to three decimal places.

Use a graphing utility to find the area of the region enclosed by the curves y = x² + 6, y = 1, x = 2, x = 5.

Evaluate cosh(cosh ^{- 1}15).

Use the Theorem of Pappus to find the volume of the solid that is generated when the region enclosed by y = 1 + x and is rotated about the x-axis.

Use cylindrical shells to find the volume of the hemisphere that results when the region in the first quadrant enclosed by the circle x² + y² = 25 is revolved about the x-axis.

Answer true or false: A right-circular cylinder of radius 8cm contains a hollow sphere of radius 7 cm. If the cylinder is filled to a height h with water and the sphere floats so that its highest point is 4cm above the water level, there is cm³ of water in the cylinder.