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