Exam 7: Applications of Definite Integrals

Find the center of mass of a thin plate of constant density covering the given region. -The region in the first and fourth quadrants enclosed by the curves $y = \frac { 4 } { 1 + x ^ { 2 } }$ and $y = \frac { - 4 } { 1 + x ^ { 2 } }$ and by the lines $x = 0$ and $x = \sqrt { 3 }$

Solve the problem. -At lift-off, a rocket weighs 40.0 tons, including the weight of fuel. It is fired vertically, and the fuel is consumed at the rate of 2.21 tons per 1,000 ft of ascent. How much work is done in lifting the rocket to an altitude of 14,000 Ft?

Solve the problem. -A tank is designed by revolving the parabola $\mathrm { y } = 5 \mathrm { x } ^ { 2 } , 0 \leq \mathrm { x } \leq 2$ , about the $\mathrm { y }$ -axis. The tank, with dimensions in meters, is filled with water weighing $9800 \mathrm {~N} / \mathrm { m } ^ { 3 }$ . How much work will it take to empty the tank by pumping the water to the tank's top? Give your answer to the nearest $\mathrm { J }$ .

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 $y = 3 x ^ { 3 }$ , below by $x$ -axis, and on the right by the line $x = 1$ , about the line $y = - 1$

Provide an appropriate response. -The region shown here is to be revolved about the $y$ -axis to generate a solid. Which of the methods (disk, washer, shell) could you use to find the volume of the solid? How many integrals would be required in each case? $y = - 2 x ^ { 2 } \quad y = - x$

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. -y = x, y = 0, x = 2, x = 4

Find the moment or center of mass of the wire, as indicated. -Find the center of mass of a wire of constant density that lies along the first-quadrant portion of the circle $x ^ { 2 } + y ^ { 2 } = 16$

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

Solve the problem. -The disk $( x - 6 ) ^ { 2 } + y ^ { 2 } \leq 1$ is revolved about the $y$ -axis to generate a torus. Find its volume. (Hint: $\int _ { - 1 } ^ { 1 } \sqrt { 1 - y ^ { 2 } } d y = \frac { 1 } { 2 } \pi$ , since it is the area of a semicircle of radius 1 .)

Find the center of mass of a thin plate of constant density covering the given region. -The region bounded by $y = x ^ { 4 } , x = 5$ , and the $x$ -axis

Find the centroid of the thin plate bounded by the graphs of the given functions. Use $\delta = 1$ and $M =$ area of the region covered by the plate. - $g ( x ) = x ^ { 2 } , \quad f ( x ) = 1$ , and $x = 0$

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

Find the centroid of the thin plate bounded by the graphs of the given functions. Use $\delta = 1$ and $M =$ area of the region covered by the plate. - $g ( x ) = x ^ { 2 }$ and $f ( x ) = x + 12$

Find the center of mass of a thin plate covering the given region with the given density function. -The region bounded by $x = y ^ { 2 }$ and the line $x = 1$ , with density $\delta ( x ) = y ^ { 2 }$

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

Solve the problem. -Find the volume that remains after a hole of radius 1 is bored through the center of a solid cylinder of radius 4 and height 8.

Provide an appropriate response. -The region shown here is to be revolved about the $y$ -axis to generate a solid. Which of the methods (disk, washer, shell) could you use to find the volume of the solid? How many integrals would be required in each case? $y = x$ $y = 4 x - x ^ { 2 }$

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the $y$ -axis $x = 5 y / 3$

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. -y = - 7x + 14, y = 7x, x = 0

Set up an integral for the length of the curve. - $x = y ^ { 2 } + 5 y , 0 \leq y \leq 5$