Exam 7: Applications of Definite Integrals

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. - $y = 9 | x | , y = 9$

Use your grapher to find the surface's area numerically. - $x ^ { 1 / 2 } + y ^ { 1 / 3 } = 3,1 \leq x \leq 4 ; x \text {-axis }$

The centroid of a triangle lies at the intersection of the triangle's medians, because it lies one-third of the way from each side towards the opposite vertex. Use this result to find the centroid of the triangle whose vertices appear as following. -(0, 0), (4, 0), (2, 8)

Find the volume of the solid generated by revolving the region about the y-axis. -The region enclosed by $x = \frac { y ^ { 2 } } { 5 } , x = 0 , y = - 5 , y = 5$

Solve the problem. -A swimming pool has a rectangular base $11 \mathrm { ft }$ long and $22 \mathrm { ft }$ wide. The sides are $5 \mathrm { ft }$ high, and the pool is full of water. How much work will it take to lower the water level 2 feet by pumping the water out over the top of the pool? Assume that the water weighs $62.4 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ . Give your answer to the nearest $\mathrm { ft } \cdot \mathrm { lb }$ .

Provide an appropriate response. -The region shown here is to be revolved about the line $x = 3$ 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 cas $x = - 4 y ^ { 2 } + 3$

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. -A 2-oz tennis ball was served at $140 \mathrm { ft } / \mathrm { sec }$ about ( $95 \mathrm { mph }$ ). How much work was done on the ball to make it go this fast? (To find the ball's mass from its weight, express the weight in pounds and divide by $32 \mathrm { ft } / \mathrm { sec } ^ { 2 }$ , the acceleration of gravity.)

Find the center of mass of a thin plate of constant density covering the given region. -The region cut from the first quadrant by the circle $x ^ { 2 } + y ^ { 2 } = 49$

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

Find the volume of the solid generated by revolving the region about the given line. -The region bounded above by the line $y = 8$ , below by the curve $y = 8 \cos ( \pi x )$ , on the left by the line $x = - 0.5$ , and on the right by the line $x = 0.5$ , about the line $y = 8$

The centroid of a triangle lies at the intersection of the triangle's medians, because it lies one-third of the way from each side towards the opposite vertex. Use this result to find the centroid of the triangle whose vertices appear as following. -(-9, 0), (9, 0), (0, 2)

Find the volume of the solid generated by revolving the shaded region about the given axis. -About the $x$ -axis $y = 3 \sqrt { \sin x }$

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. - $y = 5 x ^ { 3 } , y = 5 x \text {, for } x \geq 0$

Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. -The region bounded by $y = 7 x - x ^ { 2 }$ and $y = x$ about the line $x = 6$

Solve the problem. -A dome is in the form of a partial sphere, with a hemisphere of radius 10 feet on top and the remaining part of the sphere extending 5 feet to the ground from the center of the sphere. Find the surface area of the dome to the nearest square foot.

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

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

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 squares with bases running from the $x$ -axis to the curve.

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. - $y = 4 x ^ { 2 } , y = 4 \sqrt { x }$

Find the volume of the solid generated by revolving the region about the y-axis. -The region enclosed by $x = 2 \tan \frac { y } { 5 } , x = 0 , y = - \frac { 5 \pi } { 4 }$