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6.4 Arc Length
Length of a Curve in the Plane
If y=f(x) s a continuous first derivative on [a,b], the length of the curve from a to b is
Example:
6.5 Area of a Surface
Generated by revolving y=f(x)
about the x-axis
Generated by revolving y=f(x)
about the y-axis
Example:
6.6 Work
Generally
Work=force distance
Example
A force of 112 N is required to slide a cement block 4 m. How much work is necessary?
“work” category
6.6 Lifting
Lifting a bucket of sand Sand weighs 144 lb. Bucket weighs 4 lb. Rope weighs 0.08 lb/ft.
Bucket leaks at a steady rate and the sand is ½ gone when the bucket is lifted 18 ft.
How Much Work
a) To lift the sand?
How Much Work
b) To lift the sand and the bucket?
How Much Work
c) To lift the sand, the bucket and the rope?
As a bucket is raised 30 ft., water leaks out at a constant rate. Find the work done if the bucket originally contained 24 lb of water and 1/3 leaks out. Weight of empty bucket is 4 lb.
How Much Work
a) To lift the water?
How Much Work
a) To lift the water and bucket?
Bucket Lifted 20 ft Contains 16 lb water ( 2 gal) Leaks at constant rate Empty when reaches top
How Much Work For water only?
For water and bucket, bucket weighs 5 lb
For water and bucket and rope, rope weighs 0.08 lb/ft.
“work” category
6.6 Spring
Hooke’s Lawthe force required to stretch a spring x units beyond is “natural” length is f(x)=k x, k is the constant of the spring.
k= force length of stretch
Example
A force of 800 N stretches a spring 0.7 m.
Find the Work done.
Spring
20 N force Stretches spring 3 m beyond natural
length How Much Work
1. To stretch from natural to 5 m?
2. To stretch one additional m?
Spring
Natural length is 10 m 8 N force stretches it 1.5 m
How Much Work
1. To stretch from natural to 14 m?
2. To stretch from 11m to 13m?
Spring
250N force stretches it 30 cm
How Much Work to stretch from 20cm to 50cm?
“work” category
6.6 Pumping
Vertical cylindrical tank Diameter = 3 ft Height = 6ft Contains water, 62.5 lb/ ft^3
H M W to pump the water Of a full tank out over the top of the tank?
H M W to pump the water
Of a full tank thru a pipe which rises to a height of 4 ft above the top of the tank?
H M W to pump the water
Of a ½ full tank over the top of the tank?
H M W to pump the water
Of a ½ full tank out the pipe which rises to a height of 4 ft above the top of the tank?
Cylindrical Tank 4 m high Radius = 2m Tank buried so top is 1 m below graound
level Full of water, 1000 kg/m^3
HMW to pump full tank to ground level?
Cylindrical Tank Radius = 5 ft Height = 10 ft Full of water, 62.5 lb/ft^3
HMW to pump to a level 4 ft above ground?
Above-ground circular pool Diameter = 12 ft Height = 5 ft Depth of water is 4 ft
HMW to empty the pool by pumping the water over the top?
A open tank has the shape of a right circular cone, full of water.
Diameter of top = 8 ft, height = 6 ft
HMW to empty by pumping over the top?
A cylindrical gas tank has diameter 3 ft and is 4 ft long. The axis of the tank is horizontal.
HMW to pump the entire contents into a tractor if the opening of the tractor tank is 5 ft above the top of the tank.
“work” category
6.7 Fluid Force
is the force of a fluid against a side of a tank.
Example
Find fluid force on the vertical side of the tank in the diagram, it’s full of water (62.5).
Example
Glass tank, 3 ft. long Square ends, 1 ft. long Full of water (62.5)
Find force exerted on one end Find force exerted on one side
Example
See diagram
Right circular cylindrical tank 90 ft high 90 ft diameter Full of molasses (100 lb/ft^3)
How much force is exerted on bottom 1 ft. band?
Example
Sheet metal, area = 3 sq. ft. Submerged horizontally in 5 ft. of water
Find the fluid force on the top side.
Find the fluid force on the vertical side of a tank 3 ft. by 4 ft.
It’s full of water (62.5)
Find the fluid force on the vertical plate in the diagram..It’s submerged in water (1000 kg/cu.m.
The vertical side of a form for poured concrete is 10 ft long and 2 ft high.
Determine the force on this part of the form. Concrete weighs 140.7 lb/ft^3.
Cylindrical gas tank, horizontal axis Tank is ½ full Diameter is 3 ft. Gas weighs 42 lb/ft^3
Find fluid force on an end of the tank.
6.8 Center of Mass
6.9 Distance traveled
6.9 Position shift
6.9 Volume by slicing
6.9 Parametric graphing
6.4-6.9 Test The “key” is to know
how to use the formulae