Exam 7: Applications of Definite Integrals

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 $y = 3$ , below by the curve $y = \sqrt { 3 x }$ , and on the left by the $y$ -axis, about the line $y = 3$

Solve the problem. -A conical tank is resting on its apex. The height of the tank is $8 \mathrm { ft }$ , and the radius of its top is $7 \mathrm { ft }$ . The tank is full of gasoline weighing $45 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ . How much work will it take to pump the gasoline to the top? Give your answer to the nearest $\mathrm { ft } \cdot \mathrm { lb }$ .

Solve the problem. -A tank truck hauls oil in a 4-ft-diameter horizontal right circular cylindrical tank. If the density of the oil is $60 \mathrm { lb } / \mathrm { ft } ^ { 3 }$ , how much force does the oil exert on each end of the tank when the tank is half full?

Find the moment or center of mass of the wire, as indicated. -Find the center of mass of a wire that lies along the semicircle $y = \sqrt { 36 - x ^ { 2 } }$ if the density of the wire is $\delta =$ $| \cos \theta |$ .

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

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 $y = 1$ , below by the curve $y = \sqrt { \sin 4 x }$ , and on the left by the $y$ -axis, about the line $y = 1$

Find the volume of the solid generated by revolving the region about the y-axis. -The region enclosed by $x = y 1 / 3 , x = 0 , y = 27$

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. - $y = \sqrt { x } , y = 0 , x = 0 , x = 8$

Find the center of mass of a thin plate of constant density covering the given region. -The region bounded by $y = x ^ { 2 }$ and $y = 3$

Find the volume of the solid generated by revolving the region about the given axis. Use the shell or washer method. -The triangle with vertices (0, 0), (0, 2), and (5, 2) about the line x = 5

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 $3 x + y = 6$ , below by the $x$ -axis, and on the left by the $y$ -axis, about the line $x = - 2$

Find the length of the curve. - $y = 2 x ^ { 3 / 2 } \text { from } x = 0 \text { to } x = \frac { 5 } { 4 }$

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 center of mass of a thin plate covering the given region with the given density function. -The region between the curve $y = \frac { 8 } { x }$ and the $x$ -axis from $x = 1$ to $x = 9$ , with density $\delta ( x ) = \sqrt { x }$

Set up an integral for the length of the curve. - $y = 7 \cot x , \frac { \pi } { 4 } \leq x \leq \frac { \pi } { 2 }$

Find the length of the curve. - $x = \frac { 1 } { 3 } y ^ { 3 / 2 } - y ^ { 1 / 2 } \text { from } x = 9 \text { to } x = 16$

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

Solve the problem. -A force of 1100 lb compresses a spring from its natural length of 20 in. to a length of 12 in. How much work is done in compressing it from 12 in. to 9 in.?