1. CIVE1620- Engineering Mathematics 1.1
Lecture 10
Revolution of Curves and volume
Solid of revolution
Centroid of a plane
Centroid of a solid of revolution
Lecturer: Dr Duncan Borman

2. Revolution of the curve10 5
On a piece of paper sketch what you think the graph of
where r and h are both constants.
for,
for, r h
The path traced out by this rotation would clearly be a cylinder, of radius r and length h. This is known as a solid of revolution
Sketch what you would get if you were to:
rotate this function about the x axis, a full 3600.
This is called a revolution of the curve y=f(x).
What is the Volume of the shape?

3. r h
What about the solid of revolution produced by a general curve y = f(x) between x = a and x = b?
The volume of such a solid can be shown to be:
y = f(x) y x x y
f(x)
a b dx
Exercise - Use the integral equation above to check volume of cylinder is

4. The volume of such a solid can be shown to be:r h
The volume of such a solid can be shown to be:
Shown to be correct for a cylinder

5. The volume of such a solid can be shown to be:
Example
Consider the solid of revolution (witch’s hat shape) formed by the revolution of y=x2 about
the x­axis, for
y=x2 4
The volume of such a solid can be shown to be:

6. We can also form a solid of revolution by rotating y=f(x) about the y axis, from y=c to y=d say, and in this case the general formula for the volume is: c d dy
We need to express x (well, x2 actually) in terms of y.
This is simple for the example of y= x2, since clearly then x2 =y.
Hence the volume of the solid of revolution obtained by rotating y= x2 about the y axis between y=0 and y=4 is: x
y=x2 4
Back to our y=x2 example

7. Find the volume formed by the revolution of
about the x­axis, for

8. y y
f(x)
y = f(x)
g(x) R
y=g(x) a a b x b x

9. We can use this technique to prove the volume formula for common 3D shapes.
(see link on VLE for more information)

10. The paradox of  y  =  1/x .               
Evangelista Torricelli ( ) was a student of Galileo. 
To his amazement Torricelli's discovered an infinitely long solid with a surface that calculates to have an infinite area, but a finite volume. 
“Gabriel’s Horn”

11. Centre of mass Centroid of a plane (2D)
Centre of Mass (complex 3D shapes)
Centroid of a plane (2D)
For a plane area, the centre of mass is commonly referred to as a centroid.

12. Centroid of a plane (2D)
Suppose that the plane area we are interested in is one bounded by two known functions, y=f(x) and y=g(x) between x=a and x=b.
The co-ordinates of the centroid can then be shown to be given by:
where

13. Example
Find the centroid of the area bounded by the curve y=x2 and the line y=4 x 2 -2
We begin by determining the constant A:
Then
Then
Hence, the centroid is at:

14. Centroid of a solid (3D) yx y
In a similar way to finding the centroid of a plane area, we can find the centroid of a three-dimensional object, e.g. a solid of revolution x
A solid is formed by the rotation of y=f(x) between x=a and x=b about the x axis. Assuming the solid has constant density, the centroid will also be on the x axis (i.e. at a point where y x
a b
It remains to find the x-coordinate of the centroid, which can be shown to be given by:
where V is the volume of the solid (which found earlier)

15. Centroid of a solid (3D) - EXAMPLEy
The x-coordinate of the centroid
where V is the volume of the solid x
Earlier we found that the solid formed by the revolution of y=x2 about the x-axis
y=x2
has a volume of
Work out the integral
Earlier you found that Volume formed by the revolution of
about the x­axis, for was
Find the centroid of this volume
Hence the centroid is at

16. Earlier you found that Volume formed by the revolution of
about the x­axis, for was
Find the centroid of this volume
Work out the integral
Hence the centroid is at

17. Revolution of Curves and volume
Solid of revolution
Centroid of a plane (2D)
Centroid of a solid of revolution

18. f(x) dx Double Integrals- Volume of more complex shapes y = f(x) y x
a b dx
The volume of revolution

19. Double Integrals- Volume of more complex shapesy x
a b
In calculus of a single variable, the definite integral
is the area under the curve f(x) from x=a to x=b.
The definite integral can be extended to functions of more than one variable.
Consider a function of 2 variables z=f(x,y).
The definite integral is denoted by
        
                                  
For positive f(x,y), the definite integral is equal to the volume under the surface z=f(x,y) and above xy-plane.
See VLE links for more information

21. CIVE1620 - Engineering Mathematics 2.2
Next lecture we will look at double integrals in more detail
Final lecture we will look at Numerical methods for when it is difficult or not possible to use algebraic integration.
Often how computers, calculators etc perform very complex integration

22. CIVE1620 - Engineering Mathematics 2.2
Problem sheets
The hand in date for problem sheet 1 for CIVE1620 next week
Problem sheet will be available tomorrow .
Next lecture we are looking at Numerical methods for when it is difficult or not possible to use algebraic integration.
Often how computers, calculators etc perform very complex integration

23. What does the solid of revolution look like if the curvey
What does the solid of revolution look like if the curve
Is revolved about the y­axis? 3 x

24. x3 + k Quick Recap Integration is area under the curve
Basic integration- just reverse of differentiation (antiderivative)
More complex integration (still reverse differentiation)
We need variety of techniques to help us (more of an art to doing these)
- not fixed rules for each case as in differentiation
- needs some thinking/problem solving ability
- this is why integration can seem tough
Techniques
-Substitution
-Trig identities
-Integration by Parts
-Integration by Partial fractions

25. Integration by Substitution (change of variable)
Simplifies a function into a form we can integrate
Substitution ?

26. Techniques
-Substitution
-Trig identities
-Integration by Parts
-Integration by Partial fractions

27. Integrating trig functions
Trig Identities
1 - cos2x
sin(2x) =
2sin2x
2sin(x)cos(x)
1 + cos2x
2cos2x
cosec2x
cot2x + 1
sec2x
tan2x + 1 1
sin2x + cos2x
Last lecture we developed this to integrate more complex functions

28. Integration by parts
There is no substitution that will simplify this into a form we can integrate
What about
Integrate both sides
Try this one
Let u=x2 and dv=cos(x)

29. Integration by parts
There is no substitution that will simplify this into a form we can integrate
What about
Easier version to remember
(not as mathematically correct)
Integrate both sides
Try this one
Let u=x2 and dv=cos(x)