Project #32959 - intergration

 

Part 1: Solids of Revolution

Task 1

These are solids made by revolving a region about an axis.

(Assume that all units are in centimetres.)

 

Example: Consider the line  

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Sweep, or revolve the line f(x) about the x-axis.

What solid is formed? Think about it. Can you picture the CONE that is formed?

It has perpendicular height 8cm and radius of 4cm.

We can sketch the various elevations (or views from that side), including the dimensions.

 

                             Side and top elevation           Front elevation                      Back elevation

 

 

 

 

 

 

 

 

Using the rules listed below, find the Volume and the Surface Area of the resulting solid.                                                                            

 

   and   

 

Task 2

For each of the following functions:

(a)    Graph the function.

(b)    Describe the solid formed by revolving about the given axis.

(c)     Draw the elevations of the solid that is formed.

(d)  Calculate the Volume and Surface Area of each solid formed. (All units are in centimetres.)

                                                                       

 

  1.                                                     

Sweep the region about the x-axis.                                                                                                        

 

  1.                       

     Sweep the region about the x-axis.                                                                                                         

 


 

Part 2: Using Integration to find Volumes and Surface Areas

 

Volumes

When finding areas under curves, we divided the region up into rectangles. By making the width of the rectangles small enough, we were able to approximate the area. By letting the width, δx approach zero, we were able to find the exact area.

A similar idea is used to find Volume. Using cylinders or disks instead of rectangles, we are able to calculate the exact volumes.

Consider the following:

The volume of a cylinder or disk:

 

The volume of the thin disk P (above): 

The sum of all thin disks between a and b is approximately

Let δx0 to eliminate errors involved in using approximate cylinders, then

Volume =

                        =

                       

So, in general, the volume of a solid formed when sweeping f about the X-axis is given by :

                       

 

Surface Areas

It is a little more difficult to explain the workings behind the formula to find Surface Area so I will just give it to you.

                Surface Area:  + Area of the end bits.

Simplify this Integral fully, but if it proves to be beyond what we have done in the Maths B course you can evaluate it using the calculator. 

 

 

Task 3

In the first example of Part 2, you were asked to calculate (using basic formulae), the Volume and Surface Area of the cone created by sweeping the curve  about the X-axis.

 

Complete the following using the Integration formulae and check that the answers are the same.

 

Volume:                                                               Surface Area:  Note If y = 0.5x then y’= 0.5

              V       =                                

                        =                                         =

 

Task 4

You are now to calculate the Volume and Surface Areas using the Integral formulae of questions in Part 1, Task 2.

                                                                                                                                                           

 

Task 5

Sketch the graph defined by  

Include the co-ordinates of the end points.

Sweep the region about the X-axis to form a solid.

Show the side and front elevations of the solid.

Describe the solid as best as you can.

Calculate the Volume and the Surface Area of the solid.

 

Task 6

Sketch the graph defined by 

Include the co-ordinates of the end points.

Sweep the region about the X-axis to form a solid.

Show the side and front elevations of the solid.

Describe the solid as best as you can.

Calculate the Volume and the Surface Area of the solid.

 

 

 

Part 3: Calculate the Volume of a .303 bullet

Task 7

Pictured below is an enlargement of a .303 bullet, without it’s casing.

The diameter of the bullet is 7.90mm.

The first part of the bullet is cylindrical then it curves to a point.

The curved part could take on various forms, one of which could be the equation  

Model a variety of curves on the bullet head, decide which fits best and then calculate the volume of the bullet.

You may make full use of the graphics calculator to solve this problem.

 

 

Text Box:

 

 

 

 

 

 

 

Subject Mathematics
Due By (Pacific Time) 06/11/2014 12:00 am
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