Exam 7: Applications of Definite Integrals

Find the volume of the solid generated by revolving the region about the given line. -The region in the first quadrant bounded above by the line $y = 3$ , below by the line $y = \frac { 3 x } { 2 }$ , and on the left by the $y$ -axis, about the line $y = 3$

Solve the problem. -One end of a pool is a vertical wall $16 \mathrm { ft }$ wide. What is the force exerted on this wall by the water if it is $7 \mathrm { ft }$ deep? The density of water is $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ .

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 x-axis. - $x = 4 y ^ { 2 } , x = - 4 y , y = 4$

Find the length of the curve. - $y = \left( 16 - x ^ { 2 / 3 } \right) ^ { 3 / 2 } \text { from } x = 1 \text { to } x = 64$

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the $y$ -axis $x = 2 \tan \left( \frac { y } { 5 } \right)$

Find the length of the curve. - $y = \frac { 1 } { 6 } x ^ { 3 } + \frac { 1 } { 2 x } \text { from } x = 1 \text { to } x = 5$

Find the area of the surface generated by revolving the curve about the indicated axis. - $y = \sqrt { x } , 3 / 2 \leq x \leq 9 / 2 ; x$ -axis

Solve the problem. -A rectangular sea aquarium observation window is $14 \mathrm { ft }$ wide and $4 \mathrm { ft }$ high. What is the force on this window if the upper edge is $5 \mathrm { ft }$ below the surface of the water. The density of seawater is $64 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ .

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. - $y = \frac { 5 } { \sqrt { x } } , y = 0 , x = 1 , x = 25$

Solve. -Locate the centroid of a semicircular region.

Find the volume of the described solid. -The solid lies between planes perpendicular to the $x$ -axis at $x = \pi / 6$ to $x = \pi / 2$ . The cross sections perpendicular to the $x$ -axis are circular disks with diameters running from the curve $y = \cot x$ to the curve $y = \csc x$ .

Solve the problem. -It took 1900 J of work to stretch a spring from its natural length of 1 m to a length of 3 m. Find the spring's force constant.

Solve the problem. -Find the force on one side of a cubical container $5 \mathrm {~cm}$ on an edge if the container is filled with mercury. The density of mercury is $133 \mathrm { kN } / \mathrm { m } ^ { 3 }$ .

Solve the problem. -A vertical right circular cylindrical tank measures $26 \mathrm { ft }$ high and $10 \mathrm { ft }$ in diameter. It is full of oil weighing $60 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ . How much work does it take to pump the oil to a level $2 \mathrm { ft }$ above the top of the tank? Give your answer to the nearest $\mathrm { ft } \cdot \mathrm { lb }$ .

Find the moment or center of mass of the wire, as indicated. -Find the center of mass of a wire of constant density that lies along the line y = x from x = 0 to x = 1.

Find the center of mass of a thin plate covering the given region with the given density function. -The region bounded by the curves $y = \pm \frac { 5 } { \sqrt { x } }$ and the lines $x = 1$ and $x = 9$ , with density $\delta ( x ) = \frac { 5 } { x }$

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. - $y = x ^ { 2 } - 3 , y = 2 x , x = 0 \text {, for } x \geq 0$

Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. - $x = \int _ { 0 } ^ { y } \cos t d t , 0 \leq y \leq \pi / 3 ; y \text {-axis }$

Solve the problem. -A right triangular plate of base $4 \mathrm {~m}$ and height $2 \mathrm {~m}$ is submerged vertically, as shown below. Find the force on one side of the plate if the top vertex is $1 \mathrm {~m}$ below the surface. ( $\mathrm { w } = 9800 \mathrm {~N} / \mathrm { m } ^ { 3 }$ )

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line. -About the line $y = - 1$