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

Revolution of the curve 10 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?

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

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

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: 0 1 2

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

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

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

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

The paradox of  y  =  1/x .                Evangelista Torricelli ( 1608 - 1647 ) 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”

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.

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

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:

Centroid of a solid (3D) y x 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)

Centroid of a solid (3D) - EXAMPLE y 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

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

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

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

Double Integrals- Volume of more complex shapes y 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

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

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

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

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

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

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

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

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)

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)