Exam 7: Applications of Definite Integrals

Find the volume of the described solid. -The solid lies between planes perpendicular to the $x$ -axis at $x = - 2$ and $x = 2$ . The cross sections perpendicular to the $x$ -axis are circles whose diameters stretch from the curve $y = - 7 / \sqrt { 4 + x ^ { 2 } }$ to the curve $y = 7 / \sqrt { 4 + x ^ { 2 } }$ .

Set up an integral for the length of the curve. - $\mathrm { y } ^ { 6 } + 6 y = 6 \mathrm { x } - 1,1 \leq \mathrm { y } \leq 2$

Provide an appropriate response. -The region shown here is to be revolved about the y-axis to generate a solid. Which of the methods (disk, washer, shell) could you use to find the volume of the solid? How many integrals would be required in each case? $x = 2 y - y ^ { 2 }$

Find the fluid force exerted against the vertically submerged flat surface depicted in the diagram. Assume arbitrary units, and call the weight-density of the fluid w. -

Solve. -Find the lateral (side) surface area of the cone generated by revolving the line segment $y = x / 2,0 \leq x \leq 6$ , about the $x$ -axis.

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. -About the $\mathrm { x }$ -axis $x = y ^ { 2 } / 2$

Solve the problem. -A right triangular plate of base $8 \mathrm {~m}$ and height $4 \mathrm {~m}$ is submerged vertically, as shown below. Find the force on one side of the plate. $\left( \mathrm { w } = 9800 \mathrm {~N} / \mathrm { m } ^ { 3 } \right)$

Find the moment or center of mass of the wire, as indicated. -Find the center of mass of a wire that lies along the first-quadrant portion of the circle $x ^ { 2 } + y ^ { 2 } = 25$ if the density of the wire is $\delta = 5 \sin \theta$ .

Solve the problem. -A swimming pool has a rectangular base $11 \mathrm { ft }$ long and $22 \mathrm { ft }$ wide. The sides are $5 \mathrm { ft }$ high, and the pool is half full of water. How much work will it take to empty the pool by pumping the water out over the top of the pool? Assume that the water weighs $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ . Give your answer to the nearest $\mathrm { ft } \cdot \mathrm { lb }$ .

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line. -About the line $y = 4$

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the y-axis $x = 2 \tan \left( \frac { y } { 7 } \right)$

Find the length of the curve. - $x = \int _ { y } ^ { 1 } \sqrt { t ^ { 3 } - 1 } d t , 1 \leq y \leq 9$

Set up an integral for the length of the curve. - $\mathrm { x } = 8 \tan \mathrm { y } , 0 \leq \mathrm { y } \leq \frac { \pi } { 4 }$

Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. - $y = \cot x , 0 \leq x \leq \pi / 4 ; x$ -axis

Solve the problem. -An isosceles triangular plate is submerged vertically in seawater, with its base on the bottom. The base is $10 \mathrm { ft }$ long, and the height of the triangle is $10 \mathrm { ft }$ . Find the force exerted on one face of the plate if the water level is $2 \mathrm { ft }$ above the base of the triangle. Seawater weighs $64 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ . Round your answer to one decimal place if necessary.

Solve the problem. -The spring of a spring balance is 5.0 in. long when there is no weight on the balance, and it is 7.8 in. long with 4.0 lb hung from the balance. How much work is done in stretching it from 5.0 in. to a length of 13.3 in.?

Solve the problem. -A vertical right circular cylindrical tank measures $22 \mathrm { ft }$ high and $14 \mathrm { ft }$ in diameter. It is full of oil weighing $60 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ . How long will it take a (1/2)-horsepower (hp) motor (work output $275 \mathrm { ft } \cdot \mathrm { lb } / \mathrm { sec }$ ) to pump the oil to the level of the top of the tank? Give your answer to the nearest minute.

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. - $y = 4 \sqrt { x } , y = 0 , x = 1$

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. -About the $x$ -axis

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = \sqrt { \sin 7 x } , y = 0,0 \leq x \leq \frac { \pi } { 7 }$