Exam 7: Applications of Definite Integrals

Solve the problem. -Find a curve through the point $( - 4,1 )$ whose length integral, $1 \leq y \leq 2$ , is $L = \int _ { 1 } ^ { 2 } \sqrt { 1 + \frac { 4 } { y ^ { 3 } } } \mathrm { dy }$ .

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

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 semicircle $y = \sqrt { 49 - x ^ { 2 } }$

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 $\mathrm { x }$ -axis are circular disks whose diameters run from the parabola $y = x ^ { 2 }$ to the parabola $y = 32 - x ^ { 2 }$ .

Find the volume of the solid generated by revolving the region about the y-axis. -The region enclosed by $x = \sqrt { \sin 3 y } , 0 \leq y \leq \frac { \pi } { 6 } , x = 0$

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 region bounded by the given lines and curves about the x-axis. - $y = x ^ { 2 } , y = 16 , x = 0$

Find the volume of the solid generated by revolving the region about the given line. -The region bounded above by the line $y = 9$ , below by the curve $y = 9 - x ^ { 2 }$ , and on the right by the line $x = 3$ , about the line $y = 9$

Find the center of mass of a thin plate covering the given region with the given density function. -The triangular region cut from the first quadrant by the line $y = - x + 8$ , with density $\delta ( x ) = x$

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

Solve the problem. -A construction crane lifts a bucket of sand originally weighing 135 lb at a constant rate. Sand is lost from the bucket at a constant rate of 0.5 lb/ft. How much work is done in lifting the sand 70 ft? (Neglect the weight of The bucket.)

Find the center of mass of a thin plate of constant density covering the given region. -The region bounded by the parabola $x = y ^ { 2 }$ and the line $x = 9$

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 $y = x ^ { 3 }$ , on the right by the line $x = 2$ , and below by the $x$ -axis

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

Find the moment or center of mass of the wire, as indicated. -Find the center of mass of a wire that lies along the first-quadrant portion of the circle $x ^ { 2 } + y ^ { 2 } = 25$ if the density of the wire is $\delta = \cos \theta$ .

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

Find the moment or center of mass of the wire, as indicated. -Find the moment about the $x$ -axis 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. - $y = 3 \csc x , y = 3 \sqrt { 2 } , \frac { \pi } { 4 } \leq x \leq \frac { 3 \pi } { 4 }$

Solve the problem. -The gravitational force (in lb) of attraction between two objects is given by $F = k / x ^ { 2 }$ , where $x$ is the distance between the objects. If the objects are $5 \mathrm { ft }$ apart, find the work required to separate them until they are $50 \mathrm { ft }$ apart. Express the result in terms of $k$ .

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. - $x = 3 \sqrt { y } , x = - 3 y , y = 1$