M1M1 Treasure Hunt

The M1M1 treasure hunt, week 8

Chapter 5: Dr Hu and the dialects

Ann stormed into the blue box. "I am not at all impressed," she raged. "You ask me to do you a favour, and then you abandon me to whatever dangers might befall, without any warning."
"Ah, yes, that was a bit sneaky of me," admitted the Doctor. "But do you think you'd still have done it if you knew I had to leave you there? If I'd stayed, when you fixed the time-line of the blue cube, it would also have stopped my box from being able to travel. And I wouldn't then have been able to take you anywhere you choose as a reward," he added placatingly.
"Do you really think you can get round me that easily?" snapped Ann.
"Probably not, but I can try. My, isn't it cold today, here in London. And so wet. And windy. Do you know there's a mathematical theorem which proves that at any time, somewhere in the world there is no wind? Not here, obviously. Maybe somewhere warm, and balmy; the South of France, or the Bahamas. Or Italy, perhaps. Pisa's very nice, or at least it was a few decades ago."
"Pisa? Is that the place with the leaning tower?"
"Why yes, indeed. And a university - we had a mutual acquaintance there. Why don't we - how do you say - 'check it out'?" and with somewhat suspicious speed he set the controls, and off they went.

They opened the door to behold the famous leaning tower, swathed in the setting sun. The Doctor beamed at it proudly - "The perfect angle - one of my finest achievements," he chuckled.
"You mean it's your doing that it leans? I might have known. Doctor, why are we here? Don't pretend it's to give me a holiday in the sun."
"Of course that's the main reason! Naturally. But while we're here, I thought we might help someone we know who's studying at Pisa university. He's not an applied mathematician, but he's taking part in an engineering project involving the tower."
Just then a young student came round the corner. He muttered to himself,
"È tardi - devo finire questo calcolo oggi," as he hurried towards the Tower. Ann and the Doctor followed, waiting for the automatic translation to set in again.
"Now let's see. The length of the tower is L and its top is h above the ground. Its shape is y = x tan α, where α is constant and y is measured upwards. I need to find the integral 0L y ds. What does ds mean for heaven's sake? Maledetto professore!"
"Perhaps my colleague can be of some assistance," began Dr Hu, looking at Ann expectantly.
"I think it means an integral along a curve; s is arclength starting at the origin at the bottom of the tower," Ann explained, a little embarrassed, but of course the young man did not recognise her. "And I think the answer is
Lh." ?

"Grazie, grazie!" enthused the student. " My name's Aless. I'm a mathematician. There is a big engineering project to straighten the tower, and I've been asked to help."
"Straighten the tower?? The fools! They mustn't do that!" cried the Doctor in some agitation.
"Well, many of us agree - the leaning tower gives Pisa its distinction. But they say it's progress. Don't go away, I'll find out what we need to do next." and Aless ran off.

"Why does it matter if they straighten the tower?" asked Ann.
"You remember the story of the tower of Babel? And how it gave rise to all the languages of the world? Well, this tower is smaller and cannot create new languages. But it has given rise to terrible mutations, known as The Dialects. Dialects are evil metal monsters bent on universal domination. Last time I was here, they chased me through their lair, beneath the tower. But I managed to trap them underground by shifting the tower's foundations. OK, it's maybe a bit crooked now, but I did entomb the dialects, saving mankind. Now these pagliacci are going to free them. We must stop this. When he gets back, make sure you give him a wrong answer."

Just then, the young Italian returned. "Amici! They have given me an important task. We are going to give the foundations a kick, so that they move into a newly excavated hole and the entire tower will swing back to vertical. I have to make sure they don't use too much explosive. In particular I need to find the real values of m > 0 and n ≥ 1 for which the following integral exists:
0 (2-x-1+mx)(1-cos π x)/[x9/2(x-n)n] dx.
(m, n) = ?

"I told you to give him the wrong answer!" screamed the doctor.
"Don't worry, he always gets his m and n the wrong way round," smirked Ann knowingly. Sure enough, in a little while there came the sound of a massive explosion. The tower swung round too fast and then bounced back to its former angle, trapping most of the dialects once more.

But the drama was not yet over. During this process a single Dialect had been freed from its prison. It now emerged from the rubble and trundled tremulously down the street, calling out in a menacing metallic monotone

"Stermina! Stermina!"

"Climb the tower!" cried the Doctor to Ann and Aless as anyone scattered. "I did this once before with Galileo. The dialect'll pass directly under us, and we can drop a weight on its head. But I need to know its precise volume. Quickly, Ann, Aless! Its shape is symmetric about the x-axis. In the (x,y) plane, its round head is like y = sqrt(18x - 2x2) from x = 0 to x = 1, then y is constant until x = 3, and then y increases with gradient 1/4 until its flat bottom at x = 11."

"What a tedious calculation," moaned Aless. "This is the last time I get involved with a Mathematical Analysis of Foundations."

Dialect volume = π ?