Exam 7: Applications of Definite Integrals

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 line $x = - 1$

Solve. -Find the volume of the torus generated by revolving the circle $( x - 7 ) ^ { 2 } + y ^ { 2 } = 1$ about the $y$ -axis.

Use your grapher to find the surface's area numerically. - $y = \sin x , 0 \leq x \leq \pi / 4 ; x \text {-axis }$

Find the center of mass of a thin plate of constant density covering the given region. -The region enclosed by the parabolas $y = - x ^ { 2 } + 50$ and $y = x ^ { 2 }$

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

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

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 region bounded by the given lines and curves about the x-axis. - $y = \sqrt { 7 x } , y = 7 , x = 0$

Solve the problem. -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, $\mathrm { y }$ .

Solve the problem. -A construction crane lifts a 100-lb bucket originally containing 130 lb of sand at a constant rate. The sand leaks out at a constant rate so that there is only 65 lb of sand left when the crane reaches a height of 60 feet. How Much work is done by the crane?

Set up an integral for the length of the curve. - $x = \sin 5 y , - \pi \leq y \leq 0$

Solve the problem. -Find a curve through the point $\left( 1 , \frac { 5 } { 6 } \right)$ whose length integral, $1 \leq x \leq 2$ , is $\mathrm { L } = \int _ { 1 } ^ { 2 } \sqrt { 1 + 25 \mathrm { x } 10 } \mathrm { dx }$ .

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the $x$ -axis $y = 16 - x ^ { 2 }$

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 the circle $x ^ { 2 } + y ^ { 2 } = 25$ , on the right by the line $x = 5$ , and above by the line $y = 5$

Provide an appropriate response. -The region shown here is to be revolved about the $x$ -axis to generate a solid. Which of the methods (disk, washe shell) could you use to find the volume of the solid? How many integrals would be required in each case? $x = - 4 y ^ { 2 } + 3$

Find the length of the curve. - $x = \frac { y ^ { 4 } } { 8 } + \frac { 1 } { 4 y ^ { 2 } } \text { from } y = 1 \text { to } y = 3$

Solve the problem. -A conical tank is resting on its apex. The height of the tank is $16 \mathrm { ft }$ , and the radius of its top is $4 \mathrm { ft }$ . The tank is full of gasoline weighing $45 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ . How much work will it take to pump the gasoline to a level $16 \mathrm { ft }$ above the cone's top? Give your answer to the nearest $\mathrm { ft }$ ' $\mathrm { lb }$ .

A variable force of magnitude $F ( x )$ moves a body of mass $m$ along the $x$ -axis from $x _ { 1 }$ to $x _ { 2 }$ . The net work done by the force in moving the body from $x _ { 1 }$ to $x _ { 2 }$ is $W = \int _ { x _ { 1 } } ^ { x _ { 2 } } F ( x ) d x = \frac { 1 } { 2 } m _ { 2 } { } ^ { 2 } - \frac { 1 } { 2 } m v _ { 1 } ^ { 2 }$ , where $v _ { 1 }$ and $v _ { 2 }$ are the body's velocities at $x _ { 1 }$ and $x _ { 2 }$ . Knowing that the work done by the force equals the change in the body's kinetic energy, solve the problem. -How many foot-pounds of work does it take to throw a baseball 20 mph? A baseball weighs 5 oz, or 0.3125 lb.

Find the center of mass of a thin plate of constant density covering the given region. -The region bounded by the $x$ -axis and the curve $y = 7 \sin x , 0 \leq x \leq \pi$