Exam 6: 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. - $y = \sec x , y = \tan x , x = 0 , x = \frac { \pi } { 4 }$

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. - $y = 4 e ^ { x ^ { 2 } } , y = 0 , x = 0 , x = 1$

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 $y = \frac { 3 } { x }$ , on the right by the line $x = 3$ , and above by the line $y = 2$

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 curve $y = \sqrt { 3 x }$ , and on the left by the $y$ -axis, about the line $x = - 1$

Find the volume of the solid generated by revolving the region about the y-axis. -The region enclosed by $x = 2 \tan \frac { y } { 7 } , x = 0 , y = - \frac { 7 \pi } { 4 }$

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 x-axis. - $x = 2 \sqrt { y } , x = - 2 y , y = 1$

Find the volume that remains after a hole of radius 1 is bored through the center of a solid sphere of radius $3 .$

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 $y = x ^ { 3 }$ , on the right by the line $x = 4$ , and below by the $x$ -axis

Find the volume of the solid generated by revolving the region about the y-axis. -The region enclosed by the triangle with vertices $( 0,0 ) , ( 1,0 ) , ( 1,2 )$

Find the volume of the described solid. -The solid lies between planes perpendicular to the $x$ -axis at $x = 0$ and $x = 7$ . The cross sections perpendicular to the $x$ -axis between these planes are squares whose bases run from the parabola $y = - 2 \sqrt { x }$ to the parabola $y = 2 \sqrt { x }$

Find the volume of the described solid. -The solid lies between planes perpendicular to the $x$ -axis at $x = - 5$ and $x = 5$ . The cross sections perpendicular to the $x$ -axis are circular disks whose diameters run from the parabola $y = x ^ { 2 }$ to the parabola $y = 50 - x ^ { 2 }$ .

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 $y = \sqrt { \sin 3 x }$ , and on the left by the $y$ -axis, about the line $y = - 1$

Find the volume of the described solid. -The base of a solid is the region between the curve $y = 6 \cos x$ and the $x$ -axis from $x = 0$ to $x = \pi / 2$ . The cross sections perpendicular to the $x$ -axis are squares with bases running from the $x$ -axis to the curve.

The hemispherical bowl of radius 5 contains water to a depth 4 . Find the volume of water in the bowl.

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. - $\mathrm { x } = \frac { 1 } { 6 } \mathrm { e } ^ { 2 } , \mathrm { y } = 0 , \mathrm { x } = 0 , \mathrm { y } = 1$

Find the volume of the solid generated by revolving the region about the given line. -The region bounded above by the line $y = 16$ , below by the curve $y = 16 - x ^ { 2 }$ , and on the right by the line $x = 4$ , about the line $y = 16$

Find the volume that remains after a hole of radius 1 is bored through the center of a solid cylinder of radius 2 and height 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 x-axis. - $y = 4 x , y = 8 x , y = 4$

Find the volume of the solid generated by revolving the region about the given line. -The region bounded above by the line $y = 3$ , below by the curve $y = 3 \cos ( \pi x )$ , on the left by the line $x = - 0.5$ , and on the right by the line $x = 0.5$ , about the line $y = 3$

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line. -About the line $y = - 1$ $x = 4 y - y ^ { 2 }$