Exam 7: Applications of Definite Integrals

Find the center of mass of a thin plate of constant density covering the given region. -The region between the $x$ -axis and the curve $y = 5 \csc ^ { 2 } x , \frac { \pi } { 4 } \leq x \leq \frac { 3 \pi } { 4 }$

Solve the problem. -A frustum of a right circular cone has a height of 10 m, a base of radius 5m, and a top of radius 4m. Find its volume.

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

Solve the problem. -A rectangular swimming pool has a parabolic drain plate at the bottom of the pool. The drain plate is shaped like the region between $\mathrm { y } = \frac { 1 } { 2 } \mathrm { x } ^ { 2 }$ and the line $\mathrm { y } = \frac { 1 } { 2 }$ from $\mathrm { x } = - 1$ to $\mathrm { x } = 1$ . The pool is $10 \mathrm { ft }$ by $20 \mathrm { ft }$ and $8 \mathrm { ft }$ deep. If the pool is being filled at a rate of $200 \mathrm { ft } ^ { 3 } / \mathrm { hr }$ , what is the force on the drain plate after 4 hours of filling? Round your answer to two decimal places if necessary.

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

Solve the problem. -A vertical right circular cylindrical tank measures $20 \mathrm { ft }$ high and $12 \mathrm { ft }$ 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 the level of the top of the tank? Give your answer to the nearest $\mathrm { ft } \cdot \mathrm { lb }$ .

Provide an appropriate response. -The first-quadrant region bounded by $y = \sqrt { 4 - x ^ { 2 } }$ , the $x$ -axis, and the $y$ -axis is revolved about the $y$ -axis to form a solid. Explain how you could use elementary geometry formulas to verify the volume of the solid.

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 $y = 3 \sin \left( x ^ { 2 } \right)$

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = 9 \cos \pi x , y = 0 , x = - 0.5 , x = 0.5$

Find the fluid force exerted against the vertically submerged flat surface depicted in the diagram. Assume arbitrary units, and call the weight-density of the fluid w. -

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the $\mathrm { y }$ -axis $y = \sqrt { 2 x }$

Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. -The region in the first quadrant bounded by $x = 6 y - y ^ { 2 }$ and the $y$ -axis about the $x$ -axis

Solve the problem. -A right circular cylinder is obtained by revolving the region enclosed by the line $x = r$ , the $x$ -axis, and the line $y = h$ , about the $y$ -axis. Find the volume of the cylinder.

Find the volume of the described solid. -The solid lies between planes perpendicular to the $x$ -axis at $x = - 4$ and $x = 4$ . The cross sections perpendicular to the $x$ -axis between these planes are squares whose bases run from the semicircle $y = - \sqrt { 16 - x ^ { 2 } }$ to the semicircle $y = \sqrt { 16 - x ^ { 2 } }$ .

Solve the problem. -A water noodle is formed from a cylinder of radius 4 and height 8 by drilling through the diameter of the cylinder with a drill bit of radius 1. Find the volume of the water noodle.

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, y = 0, x = -2, x = 6

Find the area of the surface generated by revolving the curve about the indicated axis. - $x = y ^ { 3 } / 14,0 \leq y \leq 4$ ; $y$ -axis

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 = 5x, y = 0, x = 2; revolve about the line x = -3

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the $\mathrm { x }$ -axis $y = - 4 x + 8$

Find the volume of the solid generated by revolving the region about the y-axis. -The region enclosed by the triangle with vertices (1, 0), (1, 2), (3, 2)