CIVE1620- Engineering Mathematics 1.1 Lecturer: Dr Duncan Borman Integration by Partial fractions Further applications of integration - acceleration, velocity, time - mean value Lecture 9

Techniques-Substitution -Trig identities -Integration by Parts -Integration by Partial fractions Integration Partial fractions What’s this got to do with integration?

Partial fractions We’ll look at partial fractions first then come back to the integration later Quick reminder from adding fractions Put over same common denominator With partial fractions we are just doing the reverse process – “decomposing the fraction” in to it’s original parts

Partial fractions Approach (2 methods –be aware of both) - Write down equation as above - Multiply both sides of equation by denominator of LHS (x+5)(2x+1) in this case - Group terms:-Equate like terms: x:x: constants: 1 -Solve simultaneous equ: x2x - Sub back in1 Method 1 - Multiply out:

Partial fractions Approach (2 methods –be aware of both) - Write down equation as above - Multiply both sides of equation by denominator of LHS (x+5)(2x+1) in this case Set x=-½ : Method 2 Set x=-5:

Partial fractions Approach (2 methods –be aware of both) - Write down equation as above - Multiply both sides of equation by denominator of LHS (x+5)(2x+1) in this case Set x=-½ : Method 2 Set x=-5:

Partial fractions Integrate the following

Partial fractions Approach - Write down equation as above - Multiply both sides of equation by denominator - Multiply out: - Group terms: -Equate like terms: x:x: constants: 1 2 -Solve simultaneous equ: 1 2x2x 2 - Sub back in1 Try these: 1) 2)

- Multiply out: - Group terms: -Equate like terms: x:x: constants: 1 -Solve simultaneous equ: Sub back in1

Let x =2 Let x =3 Let x =-1

Sub back in4 - Multiply out: - Group terms: -Equate like terms: x:x: constants: 1 2 x2:x2: 3 -Solve simultaneous equ: x Sub back in1

Like the previous examples we can tidy up the ln terms as follows

1) Improper fractions There is a link on the VLE outlining how to do this IF NOT -Need to multiply out denominator and divide into numerator The described method only works if the degree of the numerator is less than the degree of the denominator 2) Repeated roots Extending this work on partial fractions

2) Repeated roots Extending this work on partial fractions - Multiply by denominator: Let x =1 Let x =-3 -Equate like terms: x2:x2:

Multiple choice Choose A,B,C or D for each of these: Which of these substitutions would be suitable to help integrate 1) A B C D

Multiple choice Choose A,B,C or D for each of these: Which of these substitutions is suitable to integrate 2) A B C D

Multiple choice Choose A,B,C or D for each of these: Which trig identity would I be best to start with to integrate: 3) A B C D

Multiple choice Choose A,B,C or D for each of these: If using integration by parts to integrate the following what would you use for u and dv? 4) A B C D

Multiple choice Choose A,B,C or D for each of these: We are intergrating the same thing with u=x 3 and dv= cos( x ) How many times do you anticipate having to use integration by parts to get an answer? 4) A B C D

APPLICATIONS OF INTEGRATION Physical applications The acceleration of an object as a function of time t is given by The velocity of the object is then given by Suppose that the initial velocity at time t=0 is Falling under gravity, its acceleration will be approximately constant at 9.8 m/s 2. Then k is a constant t=0 then this equation becomes t Image of Plane ©The Virtual Union 2010, sourced from Rogue-plane-attacks-and-kills-jogger-on-South-Carolina-beach-Scrape-TV-The-World-on-your-side.html Available under creative commons license Small image of tandem skydivers ©Zach Casper, sourced from Available under creative commons license

Example A ball is thrown vertically from the top of a building 80m high, and hits the ground 5 seconds later. What initial velocity was the baIl given? We have shown above that the (downward) velocity, given an initial velocity of v 0, is t=0 t At any time t, the distance s(t) below the top of the building is given by the integral, of the velocity: ( k constant) We know at time t=0, the ball has travelled no distance, it follows that k= 0, so: v 0 = -8.5 m/s. That is, the ball must be thrown upwards at an initial velocity of 8.5 m/s. We are given that at t=5 seconds, and so s=ut+ ½ at 2

Ordinary differential equations (ODE) Examples There are techniques to solve these, e.g. C= ??? or v=?? Also need initial conditions to say what the situation is say at time zero, e.g. v(0) = 10m/s

Example A 50 kg mass is shot from a cannon straight up with an initial velocity of 10m/s off a bridge that is 100 meters above the ground. If air resistance is given by 5v determine the velocity of the mass when it hits the ground. Solution m Solving the ODE to find the velocity gives:

Example A 50 kg mass is shot from a cannon straight up with an initial velocity of 10m/s off a bridge that is 100 meters above the ground. If air resistance is given by 5v determine the velocity of the mass when it hits the ground. Solution m Solving the ODE to find the velocity gives:

Example A 50 kg mass is shot from a cannon straight up with an initial velocity of 10m/s off a bridge that is 100 meters above the ground. If air resistance is given by 5v determine the velocity of the mass when it hits the ground. Solution m Solving the ODE to find the velocity gives: Finding time spent in the air, we need an equation for displacement We can use the fact that s(0) = 0 to find that c = -1080

Example A 50 kg mass is shot from a cannon straight up with an initial velocity of 10m/s off a bridge that is 100 meters above the ground. If air resistance is given by 5v determine the velocity of the mass when it hits the ground. Solution m Finding time spent in the air, we need an equation for displacement To determine when the mass hits the ground we just need to solve. which gives (solve with computer) The velocity of the object upon hitting the ground is then. l

Mean values If we wish to know the mean (average) height of N people, we know that all we need to do is add up the heights of all N people and divide by N. How do we generalise this concept to apply to a function (where N is effectively infinite)? The natural counterpart is the mean value m of a function f(x) from x=a to x=b, given by: ©Rennett Stowe 2009, sourced from Available under creative commons license

Example (mean value) Back to the falling ball. t=0t We have shown the velocity of an object falling under gravity, as a function of time t, is: This time, let us suppose that the object is initially just dropped, so the initial velocity, gives: The average velocity v m for the first 5 seconds of the object’s fall is then given by: In this simple case, because the velocity is linear, it is straightforward to work out the average velocity. If a more complex velocity is used the approach becomes much more powerful Finding an average velocity

Question (mean value) The velocity of a train (in m/s) over the first 20 seconds after leaving a station at any time t can be shown to be: Find the average velocity v m of the train during the first 20 seconds of travel ©Ian Brittany, sourced from Available under creative commons license

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

What is the integral of ln(x)?

e.g. Check by differentiating We can use integration by parts

Multiple choice Choose A,B,C or D for each of these: What has the following video got to with integration? A B C D 5) It’s a bit tenuous, but I think integration might be involved with centre of mass –that’s important in whether the car tips over. latedhttp:// lated Not much, it’s just a distraction to wake up the back row Again, tenuous, but integration can be used to work out the volume of air in the car tyres Integration can be used to calculate workdone, so could be used to calculate energy required to get car up to the slope (if we knew the equation of the slope and frictional losses etc)

CIVE Engineering Mathematics 2.2 I Next lecture we are looking at 3D integration to find volumes and centre of mass Techniques-Substitution -Trig identities -Integration by Parts -Integration by Partial fractions

Question (mean value) The velocity of a train over the first 20 seconds after leaving a station at any time t can be shown to be: Find the average velocity v m of the train during the first 20 seconds of travel use the substitution