More Applications of the Integral surface areas, volumes and more…

More Applications of the Integral

surface areas, volumes and more…

Mathematics and Architecture

• Mathematical forms have always played a huge role in architecture and Calculus is a critical mathematical tool in architectural design and civil engineering.

• The wonderful shapes of Frank Lloyd Wright’s Guggenheim Museum or the Support columns of the Johnson Wax building evoke strong mathematical ideas.

Frank Lloyd Wright 1867 – 1959

Interior of the Guggenheim Museum, New York

• Example: What’s the volume of a pyramid?

– Let the height of the pyramid be ‘h’ and add together a series of slabs

– We need to devise a strategy to relate ‘a’,’b’ and ‘h’

– We need to set up a Riemann Sum that represents this

– Finally we need to go from a “frustrum” to the entire pyramid

Not all volumes are made by revolution

• What is the volume produced by the intersection of two cylinders of radius ‘R’?

– What does the common intersection look like?

– How can we turn this in a Riemann Sum?

This will really tie you in knots!

• Start with a diagram showing the intersection a bit clearer

• Slice the cylinders along their “long” axes – what do you see?

• Repeat this but this time moving toward the top – again what shape is the common intersection?

A deeper look at integration…

• Integration provides more than volumes and areas. At a deeper level the value of an integral is a measure of the total change in a process. Consider a simple example – accelerated motion:

• A car is moving with a velocity given by the expression:

You may have seen the solution expresed this way:

2( ) 2v t t

2

2

2

( ) 2

2

2

dxv t t

dt

dx t dt

x t dt

A deeper look at integration…

• The previous method is “perfectly correct” but not very instructive. Consider the same problem from a “Riemann Sum” perspective:

• What meaning could you give to • Express the total distance traveled as a Riemann Sum

and show that this is the same result that we got before.

( )i ix v t t

Take home message: Integration “puts together” all the small changes that

occur in a process and gives us the total change.

The Strange Disappearance of the USS Enterprise!

• President I have some good news and some bad news. The good news first - we have finally gotten rid of Acting-Ensign Wesley Crusher. I knew you'd be pleased. Now the bad news - we had to lose the entire Enterprise to do this!

It seems that the USS Enterprise got too close to a black hole or some energy field or whatever (probably one of Crusher's dopey experiments!). Any way, the Enterprise started to break up and get sucked in. The black hole's mass began to grow at a rate described by the function

2( )10 tdm tte

dt

where t was measured in minutes and dm/dt is in units of millions of kg per minute.

It’s not pretty but the graph of the event looks like this!

The president looked puzzled and then asked:

1. What does the slopeof this graph tell you?

2. The total mass of the ship was 4 million kg. How long did it take for her to go?

3. On average, how quickly didshe (the Enterprise) go?

"Is that all there is" asked a pensive president.No President, I have this for you wrapped in the flag. Its the only piece of the Enterprise that didn't get sucked in.With reverence the president un-wrapped the flag. Inside she saw:

A Star Mangled Spanner!

Average Value of a Function (section 5.4 pg304)

What is the average value of a continuously changing function? For exampleconsider …

22 , (0,2)y x x x

We define an average for a function this way…

( )( )

b

aaverage

f x dxf x

b a

( ) averageb a f ( )b

a

f x dx

Try pg 305 #8,11

Wrapping up sections

• Make sure you understand the basic idea of what a Riemann sum is. Be prepared to explain this to me in a paragraph or two!

• You will not be examined on any of the volume, area or arc length methods that we did not formally discuss in class (but you are expected to be able to use these on assignments if necessary)

• Don’t lose sight of the basic idea of what integration is all about.

Surface Area (to be continued later)

• Develop an expression for the surface area of the surface formed by rotating y = 1/x around the x-axis between x = 1 and x = 10

• Generalize for all functions f(x), show that:

22 ( ) 1 '( )b

a

SA f x f x dx

Surface Area

• Devise a method to find the surface area of the paraboloid formed by revolving y = x2 around the y-axis between x = 0 and x = 3. This should be integrable so find the final answer.

Be sure to read pgs 488-489