Exam 6: Applications of Integration

A flat plate of negligible thickness is in the shape of a right triangle with base 5 ft and height 10 ft. The plate is submerged in a tank of water. Find the force on the face of the plate under the following conditions: (Use 62.4 lb/ft³ as the density of water.) (a) The plate is submerged horizontally so that it rests flat on the bottom of the tank at a depth of 14 ft.(b) The plate is submerged vertically, base edge up and base at a depth of 3 ft.

Find the total hydraulic force on a dam in the shape of an equilateral triangle with one vertex pointing down, if the side of the triangle is 100 feet and the water is even with the top.

Find the length of the curve $x = \frac { 1 } { 8 } y ^ { 4 } + \frac { 1 } { 4 } y ^ { - 2 } , \quad 1 \leq y \leq 2$

How much work is done to bring a load of ore to the surface if a miner uses a cable weighing 2 lb/ft to haul a 100 lb bucket of ore up a mine shaft 800 feet deep?

If the demand function for a certain commodity is $p ( x ) = 4 - \frac { x } { 30 }$ and the consumer surplus is 15, what should be the sale price?

Suppose a hemispherical tank with radius 10 feet is filled with liquid whose density is 62 pounds per cubic foot. Find the work required to pump all the liquid out through the top of the tank.

Find the area of the region bounded by $x = 1 + \sin t , y = \sin t , 0 \leq t \leq \pi$ , and the x-axis.

A rental estate management company manages an apartment complex with 20 units. The manager estimates that all 20 units can be rented if the rent is $150 per unit per month and that for each increase in rent of $10, one apartment will be vacated. Find the consumer surplus when 5 apartments are vacated.

A 100-foot length of steel chain weighing 5 lb/ft is dangling from the drum of a winch.(a) How much work is required to wind the chain onto the drum? (b) How much work is required to wind the first 30 feet of chain onto the drum?

The temperature (in °F) in a certain city t hours after 9 A.M. is approximated by the function $T ( t ) = 50 + 14 \sin \left( \frac { \pi } { 12 } t \right)$ . Find the average temperature during the period from 9 A.M. to 9 P.M..

The marginal cost for the production of the new Super Widget at Widget International is give by $C ^ { \prime } ( q ) = 15 - 0.0002 q$ , whereas the marginal revenue is $R ^ { \prime } ( q ) = 20 - 0.003 q$ , where in both cases q represents the number of units produced. (a) Determine the change in profits when sales are increased from 700 to 1700 units. (b) What is the change in profit when sales increase from 1700 to 2700 units? Discuss your answer.

The base of a certain solid is the triangular region with vertices (0,0), (1,1), and (2,0). Cross-sections perpendicular to the x-axis are semicircles. Find the volume of the solid.

Find the length of the curve $x = t ^ { 3 } - 3 t ^ { 2 } , y = t ^ { 3 } + 3 t ^ { 2 } , 0 \leq t \leq \sqrt { 5 }$ .

Find the length of the curve $y = f ( x ) , 0 \leq x \leq 2 , \text { if } f ^ { \prime } ( x ) = \sqrt { 4 x ^ { 2 } + 2 x - \frac { 3 } { 4 } }$

Find the area of the region bounded by the curves $f ( x ) = \frac { 7 } { 2 } x + 1$ and $g ( x ) = 2 ^ { x }$ .

Calculate the center of mass of a lamina with the given density and shape:

Find the area of the region bounded by the curves $x = 4 - y ^ { 2 }$ and $x = y ^ { 2 } - 4$ .

A force of 20 pounds stretches a spring $\frac { 3 } { 4 }$ foot beyond its natural length. Find the work done in stretching the spring 1 foot.

If the demand function for a certain commodity is $p ( x ) = 4 - \frac { x } { 30 }$ and the consumer surplus is 15, what should be the production level?

Find the center of mass of the lamina of uniform density $\delta$ bounded by $y = \sqrt { 1 - x ^ { 2 } }$ and x-axis.