Exam 7: Applications of Definite Integrals

Solve the problem. - $\text { How much work is done by a } 120 - \mathrm { lb } \text { woman as she walks up } 8 \text { steps, each with a } \frac { 1 } { 2 } - \mathrm { ft } \text { rise? }$

Set up an integral for the area of the surface generated by revolving the given curve about the indicated axis. - $x y = 4,2 \leq y \leq 3 ; y$ -axis

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. - $y = 6 x ^ { 3 } , y = 6 x$ , for $x \geq 0$

Find the center of mass of a thin plate covering the given region with the given density function. -The region bounded by the parabola $y = 64 - x ^ { 2 }$ and the $x$ -axis, with density $\delta ( x ) = 8 x ^ { 2 }$

Solve the problem. -Find the volume that remains after a hole of radius 1 is bored through the center of a solid sphere of radius 2. Round to the nearest tenth.

Solve the problem. -Find a curve through the point $( 0,3 )$ whose length integral, $0 \leq x \leq 1$ , is $L = \int _ { 0 } ^ { 1 } \sqrt { 1 + 4 x ^ { 2 } } \mathrm { dx }$ .

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 $x = 3 y - y ^ { 2 }$ and the $y$ -axis about the $y$ -axis

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves about the given lines. - $y = 3 x , \quad y = x ^ { 2 }$ ; revolve about the $y$ -axis

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated line. -About the line $y = - 1$ $x = 6 y - y ^ { 2 }$

Find the volume of the described solid. -The base of a solid is the region between the curve $y = 5 \cos x$ and the $x$ -axis from $x = 0$ to $x = \pi / 2$ . The cross sections perpendicular to the $x$ -axis are isosceles right triangles with one leg on the base of the solid.

Solve the problem. -A bead is formed from a sphere of radius 3 by drilling through a diameter of the sphere with a drill bit of radius 1. Find the volume of the bead.

Find the moment or center of mass of the wire, as indicated. -Find the moment about the $y$ -axis of a wire of constant density that lies along the curve $y = x ^ { 2 }$ from $x = 0$ to $x = \sqrt { 2 }$

Find the center of mass of a thin plate of constant density covering the given region. -The region between the curve $y = \frac { 4 } { \sqrt { x } }$ and the $x$ -axis from $x = 1$ to $x = 16$

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

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. - $y = 8 x , y = - \frac { x } { 8 } , x = 1$

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 $y = 2$ , below by the curve $y = \sqrt { 2 x }$ , and on the left by the $y$ -axis, about the line $x = - 1$

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves about the given lines. - $y = 5 x , \quad y = 0 , \quad x = 3$ ; revolve about the $x$ -axis

Find the volume of the solid generated by revolving the region about the y-axis. -The region enclosed by the triangle with vertices (0, 0), (4, 0), (4, 2)

Solve the problem. -An auxiliary fuel tank for a helicopter is shaped like the surface generated by revolving the curve $y = 1 - \frac { x ^ { 2 } } { 16 } , - 4$ $\leq x \leq 4$ , about the $x$ -axis (dimensions are in feet). How many cubic feet of fuel will the tank hold to the nearest cubic foot?

Solve the problem. -A spring has a natural length of 26 in. A force of 1600 lb stretches the spring to 36 in. How far beyond its natural length will a 400 lb force stretch the spring?