Exam 7: Area of a Region Between Two Curves

Thr area of the top side of a piece of sheet metal is 9 square feet. The sheet metal is submerged horizontally in 8 feet of water. Find the fluid force on the top side. Round your answer to One decimal place.

A tank with a base of 4 feet by 3 feet and a height of 4 feet is full of water. The water weighs 62.4 pounds per cubic foot. How much work is done in pumping water out over the top edge In order to empty half of the tank. Round your answer to one decimal place.

Find the center of mass of the given system of point masses. 10 1 9 3 , (-8,8) (1,8) (2,-10) (6,6)

Find the area of the region bounded by the graphs of the equations. $f ( x ) = \frac { 18 x } { x ^ { 2 } + 1 } , y = 0,0 \leq x \leq 9$

Neglecting air resistance and the weight of the propellant, determine the work done in propelling a six-ton satellite to a height of 200 miles above earth. Assume that Earth has a radius of 4000 miles.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line $y = 14$ $y = \sin x , y = 0,0 \leq x \leq \frac { \pi } { 2 }$

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the x-axis. $y = x ^ { 6 } , x = 0 , y = 64$

Find the fluid force on the vertical side of the tank, where the dimensions are given in feet. Assume that the tank is full of water. Note: The density of water is 62.4 lbs per cubic foot.

Use the shell method to set up and evaluate the integral $y = 3 \sqrt { x } \text { that gives the }$ volume of the solid generated by revolving the plane region about the y-axis.

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis. $y = 14 x - x ^ { 2 } , x = 0 , y = 49$

Find the area of the surface generated by revolving the curve about the y-axis. $y = \sqrt [ 3 ] { x } + 10,1 \leq x \leq 1,000$

The figure is the vertical side of a form for poured concrete that weighs 140.7 pounds per cubic foot. Dimensions in the figure are in feet. Determine force on this part of the concrete form

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the $x \text {-axis. }$ -axis. Verify your results using the integration capabilities of a Graphing utility. $y = \sin ( x ) , y = 0 , x = 0 , x = \frac { \pi } { 3 }$

Electrical wires suspended between two towers form a catenary modeled by the equation $y = 20 \cosh \frac { x } { 20 } , - 25 \leq x \leq 25$ where $x$ and $y$ are measured in meters. The towers are 50 meters apart. Find the length of the suspended cable. Round your answer to three decimal places.

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region bounded by $y = \sqrt { x + 20 } , y = x , y = 0$ about the $x$ -axis.

Use the disk or shell method to find the volume of the solid generated by revolving the region in the first quadrant bounded by the graph of the equation about the given line. $x ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } = 7 ^ { \frac { 2 } { 3 } }$ (i) the $x$ -axis; (ii) the $y$ -axis

A cylindrical gasoline tank 4 feet in diameter and 4 feet long is carried on the bank of a truck and is used to fuel tractors. The axis of the tank is horizontal. The opening on the tractor tank Is 5 feet above the top of the tank in the truck. Find the work done in pumping the entire gasoline Contents that weighs 42 pounds per cubic feet of the fuel tank into a tractor. (Hint: Evaluate one Integral by geometric formula and the other by observing that the integrand is an odd function.)

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations $y = e ^ { x / 13 } + e ^ { - x / 13 } , y = 0 , x = - 1$ , and $x = 2$ about the $x$ -axis. Round your answer to four decimal places.

$\text { A solid is generated by revolving the region bounded by } y = \sqrt { 100 - x ^ { 2 } } \text { and }$ $y = 0$ about the $y$ -axis. A hole, centered along the axis of revolution, is drilled through this solid so that one-third of the volume is removed. Find the diameter of the hole. Round your answer to three decimal places.

Set up and evaluate the definite integral for the area of the surface formed by revolving the graph of $y = 9 - x ^ { 2 }$ about the $y$ -axis. Round your answer to three decimal places.