Exam 6: 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 $y = 7 / \sqrt { 4 + x ^ { 2 } }$ .

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the $x$ -axis $y = - 5 x + 10$

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves about the given lines. - $y = 16 - x ^ { 2 } , \quad y = 16 , \quad x = 4 ;$ revolve about the line $y = 16$

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. - $x = 3 y - y ^ { 2 } , x = 0$

Find the volume of the solid generated by revolving the region about the y-axis. -The region enclosed by the triangle with vertices $( 3,0 ) , ( 3,2 ) , ( 5,2 )$

Find the volume of the solid generated by revolving the region about the y-axis. -The region enclosed by $x = y 1 / 3 , x = 0 , y = 27$

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the $y$ -axis $x = 5 y / 6$

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

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 $y = 4 \sin \left( x ^ { 2 } \right)$

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 $y = 3 x - x ^ { 2 }$

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. - $y = 3 x ^ { 2 } , y = 3 \sqrt { x }$

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

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = \frac { 7 } { x } , y = - x + 8$

Find the volume of the described solid. -The solid lies between planes perpendicular to the $x$ -axis at $x = - 3$ and $x = 3$ . The cross sections perpendicular to the $x$ -axis are semicircles whose diameters run from $y = - \sqrt { 9 - x ^ { 2 } }$ to $y = \sqrt { 9 - x ^ { 2 } }$ .

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 $y = \sqrt { 9 - x ^ { 2 } }$

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

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 isosceles right triangles with one leg on the base of the solid.

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = 3 \csc x , y = 3 \sqrt { 2 } , \frac { \pi } { 4 } \leq x \leq \frac { 3 \pi } { 4 }$

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = \sqrt { 2 x + 3 } , y = 0 , x = 0 , x = 1$

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