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This strategy is by no means optimal nor exhaustive. It is for students who are struggling with basic integration and anti-differentiation and need something to help them start calculating straightforward integrals and finding anti-derivatives.

TL;DR: The strategy to antidifferentiate a function $f$ that I present is as follows:

1. Direct
2. Manipulation
3. $u$-Substitution
4. Parts

Please find pdf files containing solutions to the summer exam here and here.

Due to not having Adobe Acrobat in the lab and not being very good at using the scanner these files are mixed up and by and large comprise the odd pages and the even pages.

First of all results are down the bottom. The projects were all marked out of 75 and the second column is the CA mark you are carrying with Wednesday’s exam. You are identified by the last five digits of your student number. At the bottom there are some average scores.

Students with no score either handed in no project or handed up late.

The *S/N mark denotes that you were certified absent from Test 1. Your exam will now be worth 87.5.

If you would like to discuss your project please email me. A lot of people made very fundamental errors and this is where most marks were lost.

First of all results are down the bottom. You are identified by the last five digits of your student number. The scores are itemized as you can see. At the bottom there are some average scores.

If you would like to see your paper or have it discussed please email me.

Students with no score were either absent in which case they score zero — or certified absent in which case their marks carry forward to the summer exam.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jippo@campus.ie and I will add you to the mailing list.

## This Week

We finished the chapter on area and volume and we have began the chapter on differential equations.

In tutorials we looked at p. 59 Q. 1(a), 2(a)(c)(f), 3(c) and p.64 Q. 4(e)(f).

### Mistakes in Notes!!

1. The formula for the volume of a doughnut/torus on p.97 is correct — there is a $\pi^2$ term.
2. Autumn 2011 Q.2(b)(ii) P. 99 My geometric intuition led me astray here I am afraid. The equation that governs the rotation about the $x$-axis (and the generalisation to rotations about $y=d$) use a summation of cylinders and the “big volume” – “hole” idea works perfectly. However, when rotating around the $y$-axis things are different and we derived our formula using cylindrical shells (animation). This means that the “big volume” – “hole” perspective is not going to work. What is actually happening in this question is we are generating a cylindrical shell as per the animation above by rotating the below strip about the $y$-axis.

In this case the height of the strip (corresponding to height of the red plus the yellow) is given by $x-(x^2-2x)$. This is directly analogous to the situation where we find the area between two curves as the integral of “top curve minus bottom” curve. Therefore I should have never put the $-$ in front of

$\displaystyle V=\int_0^32\pi x[x-(x^2-2x)]\,dx,$

and the answer is $\frac{27}{2}\pi$. Note that we shouldn’t blindly change the $-\frac{27}{2}\pi$ that we got in class. If we use our formulae correctly then we shouldn’t get meaningless answers. In fact this question serves a timely reminder that to use a “formula” with total aptitude you should probably know how to derive it — this suggests the extent and limits of its application.

Thus, if we ever want to find the volume generated by rotating a region $B$, lying in the right-half plane, (bounded by curves $y=f(x)$ and $y=g(x)$), about the $y$-axis, we should use:

$\displaystyle V=\int_a^b 2\pi x[\text{top curve''- bottom curve''}]\,dx$.

### Exercise from Notes

Evaluate the integrals in Summer 2011 Q. 2(c)

## Test Results

Sunday all going well.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jippo@campus.ie and I will add you to the mailing list.

## This Week

On Wednesday we had our test and on Thursday we finished section 4.2 and have gotten as far as the second example in section 4.3.

In the tutorial we did p.48 Q. 2-4 and p. 59 Q. 3(a). We also spoke about the following integrals:

• $\displaystyle\int \,dx$
• $\displaystyle\int 1\,dx$
• $\displaystyle\int x^0\,dx$
• $\displaystyle\int k\,dx$
• $\displaystyle\int 0\,dx$

### Exercise from Notes

Redo Example 2 from section 4.2 using the formula we derived in section 4.3. Don’t make the same mistake I did in deriving this second formula — this needs a bit more care…

## Test Results

I won’t have them as fast as I’d like is all I can say. I’ll do my best.

## Project

Now that the test is over you may begin thinking about your project/homework. You have a choice of six projects — one for each full chapter. The final date for submission is 24 April 2012. You have full freedom in which one you want to do and can hand up early if you want. Please submit to the big box at the School of Mathematical Science.  If I were you I would aim to get it done and dusted early as this is creeping into your study time and is very close to the summer examinations.

Note that you are free to collaborate with each other and use references but this must be indicated on your hand-up in a declaration. Evidence of copying or plagiarism will result in divided marks or no marks respectively. You will not receive diminished marks for declared collaboration or referencing although I demand originality of presentation. If you have a problem interpreting any question feel free to approach me, comment on the webpage or email.

Ensure to put your name, student number, module code (MS 2002), my name, and your declaration on your homework.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jippo@campus.ie and I will add you to the mailing list.

## This Week

In lectures, we finished section 3.3. We started chapter 4 and have completed section 4.1 and are in section 4.2.

## Reminder

Test on 9 am on Wednesday in WGB G 05 — NOT not the ordinary lecture location.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jippo@campus.ie and I will add you to the mailing list.

## This Week

In lectures, we finished section 3.2 and are in section 3.3.

In tutorials we did p.42 Q. 3 and p.48 Q. 1,5 & 7

## Reminder

Test on 9 am on Wednesday February 22 in WGB G 05 — NOT not the ordinary lecture location.

## Problems

### From the Class

Show that for all non-zero $a\in\mathbb{R}$,

$\displaystyle \int e^{ax}\,dx=\frac{1}{a}e^{ax}+C$.

Nothing fancy here really just eight integrals to be taken from the exercises on page 23, 32, 36, 39, 42, 44, 48 and 59. The only thing that’ll change is the constants will be different for Tests A and B. Please find your sample here.

I just noticed this says Test A rather than Sample at the top. It is a sample and not what the actual Test A is going to be. Also I will attach a copy of the tables.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jippo@campus.ie and I will add you to the mailing list.

## This Week

In lectures, we started chapter 3 and are in section 3.2. We also looked at some simpler proofs of Theorems 1.3.2 and 3.1.4. These are reproduced below.

In tutorials we did p.36 Q 6, p.42 Q.2, 4 and  p.44 Q.1 (a)(e), 2(a).

## Sample Test

Hopefully by Tuesday morning.

Here we give an alternative proof of Theorem 3.1.4. It is very likely that you will be asked to prove some of Theorem 3.1.4 in the exam and you are free to choose which proof you prefer. You may even prove parts (ii) and (iii) of Theorem 3.1.4 using the proof in the notes and then use this proof for part (iv). In class I also gave an easier proof of Theorem 1.3.2 although this won’t be examinable.

### Theorem 3.1.4

For all positive $a$ and $b$, and any rational number $r$, we have

1. $\ln 1=0$,
2. $\ln(ab)=\ln a+\ln b$,
3. $\displaystyle\ln\left(\frac{a}{b}\right)=\ln a-\ln b$,
4. $\ln a^r=r\ln a$.