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

Find the area of the surface generated when from y = 1 to y = 2 is rotated around the y-axis.

Answer true or false. If a square, flat surface is suspended vertically in water and its center is 64 m deep, the force on the bottom side of the object will triple if the object is relocated to a depth of 192 m. Neglect the effect of the atmosphere above the liquid.

A right triangle is submerged vertically with one side parallel to the surface in a liquid of density $\rho$ . The triangle has a leg that is 12 m long located at the surface and a leg 12 m long straight down. Find the force exerted on the triangular surface, in terms of the density. Neglect the effect of the atmosphere above the liquid.

A cylindrical tank of radius 8 m and height 18 m is filled with a liquid whose density is 2.06 kg/m³. How much work is needed to lift the liquid out of the tank?

Approximate the area of the region that lies below y = cos x and above y = 0.12x, where 0 $\le$ x $\le$ $\pi$ . Approximate area to three decimal places.

Find the arc length of the curve x = y⁶ over the interval y = -2 to y = 2. Approximate area to three decimal places. Use a CAS if necessary.

Find the area of the surface of a sphere of radius r which is generated by rotating a semicircle about a diameter.

A cylindrical tank of radius 8 m and height 18 m is filled with a liquid whose density is 3.16 kg/m³. How much work is needed to lift the liquid out of the tank?

Find if y = sinh(5x²).

Find the area of the surface generated when y = x³ from x = 0 to x = 1 is rotated around the x-axis.

A force of 20x² lb is applied to an object as it moves from x = 0 ft to x = 2 ft. How much work is done?

Assume that 11 ft·lb of work is required to stretch a spring 2 ft beyond its natural length. What is the spring constant?

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

Evaluate cosh(6 sinh ^{- 1}8).

Use a graphing utility to find the area of the region enclosed by the curves y = x³, y = 0, x = -2, x = 2.

Use a graphing utility to find the area of the region enclosed by the curves y = sin (4x), y = sin (6x), x = 0, x = $\pi$ /2. Approximate area to three decimal places.

Evaluate cosh(sinh ^{- 1}24).

Find the area of the surface that is generated by revolving the portion of the curve y = x³ between x = 0 and x = 2 about the y-axis. Approximate area to three decimal places.

Answer true or false. = .

Answer true or false. If a submerged horizontal object is elevated to twice its original depth, the force exerted on the top of the object will be twice the force originally exerted on the object. Assume there is a vacuum above the liquid surface.