Exam 7: Applications of Definite Integrals

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = \sqrt { \sin 7 x } , y = 1 , x = 0 \text { to } x = \frac { \pi } { 14 }$

Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. -The region bounded by $y = 2 \sqrt { x } , y = 2$ , and $x = 0$ about the $y$ -axis

Find the volume of the solid generated by revolving the region about the y-axis. -The region in the first quadrant bounded on the left by $y = \frac { 5 } { x }$ , on the right by the line $x = 5$ , and above by the line $y = 2$

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = \sqrt { 81 - x ^ { 2 } } , y = 0 , x = 0 , x = 9$

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = x ^ { 2 } + 5 , y = 3 x + 5$

Solve the problem. -A bathroom scale is compressed $\frac { 1 } { 4 }$ in. when a $220 \mathrm { lb }$ person stands on it. Assuming that the scale behaves like a spring that obeys Hooke's law, how much does someone who compresses the scale $\frac { 1 } { 8 }$ in. weigh?

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line. -About the line $y = - 6$ x=y+6 x=

Find the volume of the described solid. -The solid lies between planes perpendicular to the $x$ -axis at $x = - 6$ and $x = 6$ . The cross sections perpendicular to the $x$ -axis are semicircles whose diameters run from $y = - \sqrt { 36 - x ^ { 2 } }$ to $y = \sqrt { 36 - x ^ { 2 } }$ .

Solve. -Find the surface area of the cone frustum generated by revolving the line segment $y = ( x / 3 ) + ( 1 / 3 )$ , $1 \leq x \leq 5$ , about the $x$ -axis.

Solve the problem. -An auxiliary fuel tank for a helicopter is shaped like the surface generated by revolving the curve $y = 1 - \frac { x ^ { 2 } } { 4 }$ , $- 2$ $\leq x \leq 2$ , about the $x$ -axis (dimensions are in feet). If a cubic foot holds $7.481$ gallons and the helicopter gets 3 miles to the gallon, how many additional miles will the helicopter be able to fly once the tank is installed (to the nearest mile)?

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. - $y = x ^ { 2 } , y = 5 + 4 x$ , for $x \geq 0$

Solve the problem. -A semicircular plate 18 ft in diameter sticks straight down into fresh water with the diameter along the surface. Find the force exerted by the water on one side of the plate.

Set up an integral for the length of the curve. - $y = \sqrt { 1 - x ^ { 8 } } , - \frac { 1 } { 4 } \leq x \leq \frac { 1 } { 4 }$

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line. -About the line $y = 5$ $x = 3 y - y ^ { 2 }$

Solve the problem. -Find the work done in winding up a 175-ft cable that weighs 3.00 lb/ft.

Find the moment or center of mass of the wire, as indicated. -Find the center of mass of a wire that lies along the semicircle $y = \sqrt { 4 - x ^ { 2 } }$ if the density of the wire is $\delta = 2 \sin \theta$ .

Solve the problem. -A spherical tank of water has a radius of $13 \mathrm { ft }$ , with the center of the tank $50 \mathrm { ft }$ above the ground. How much work will it take to fill the tank by pumping water up from ground level? Assume the water weighs $62.4 \mathrm { lb } / \mathrm { ft } 3$ . Give your answer to the nearest $\mathrm { ft } \cdot \mathrm { lb }$ .

Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. -The region bounded by $y = 3 x - x ^ { 2 }$ and $y = x$ about the $y$ -axis

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 $y = 1$ , below by the curve $y = \sqrt { \sin 2 x }$ , and on the left by the $y$ -axis, about the line $y = - 1$

Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. -The region bounded by $y = 5 \sqrt { x } , y = 5$ , and $x = 0$ about the line $y = 5$