Exam 6: Applications of Definite Integrals

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

Find the volume of the solid generated by revolving the region about the y-axis. -The region enclosed by $x = \frac { 4 } { y } , x = 0 , y = 1 , y = 2$

A water tank is formed by revolving the curve $y = 2 x ^ { 4 }$ about the $y$ -axis. Find the volume of water in the tank as a function of the water depth, $y$ .

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 $\mathrm { x }$ -axis are circular disks with diameters running from the curve $\mathrm { y } = \cot \mathrm { x }$ to the curve $\mathrm { y } = \csc \mathrm { x }$ .

An auxiliary fuel tank for a helicopter is shaped like the surface generated by revolving the curve $y = 1 - \frac { x ^ { 2 } } { 16 }$ , $= 4$ $\leq x \leq 4$ , 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 x-axis. - $x = 4 e ^ { - y ^ { 2 } } , y = 0 , x = 0 , y = 1$

Find the volume of the solid generated by revolving the region about the y-axis. -The region enclosed by $x = \sqrt { \sin 3 y } , 0 \leq y \leq \frac { \pi } { 6 } , x = 0$

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the $x$ -axis $y = 4 \sec x$

Find the volume of the described solid. -The base of a solid is the region between the curve $y = 3 \cos x$ and the $x$ -axis from $x = 0$ to $x = \pi / 2$ . The cross sections perpendicular to the $x$ -axis are squares with diagonals running from the $x$ -axis to the curve.

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = \sqrt { 25 - x ^ { 2 } } , y = 0 , x = 0 , x = 5$

A water tank is formed by revolving the curve $y = 3 x ^ { 4 }$ about the $y$ -axis. Water drains from the tank through a small hole in the bottom of the tank. At what constant rate does the water level, $y$ , fall? (Use Torricelli's Law: $\mathrm { dV } / \mathrm { dt } = - \mathrm { m } \sqrt { \mathrm { y } }$ .)

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = \sqrt { \sin 3 x } , y = 0,0 \leq x \leq \frac { \pi } { 3 }$

Find the volume of the solid generated by revolving the region about the given line. -The region in the second quadrant bounded above by the curve $y = 16 - x ^ { 2 }$ , below by the $x$ -axis, and on the right by the $y$ -axis, about the line $x = 1$

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. - $y = 5 e ^ { - x ^ { 2 } } , y = 0 , x = 0 , x = 1$

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. -About the $y$ -axis $y = \frac { 2 \sin ( x ) } { x } ; 0 < x \leq \pi$

Find the volume of the described solid. -The base of the solid is the disk $x ^ { 2 } + y ^ { 2 } \leq 4$ . The cross sections by planes perpendicular to the $y$ -axis between $y = - 2$ and $y = 2$ are isosceles right triangles with one leg in the disk.

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

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 volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = 8 \csc x , y = 0 , x = \frac { \pi } { 4 } , x = \frac { 3 \pi } { 4 }$

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