# First Lecture

It’s was great seeing you this morning, and the attendance (esp. early birds) are really encouraging for me. So let me repeat what I said this morning here: if I’m still seeing so many early birds (like before 9:15), I’ll try to arrive every time by 9:15am and we can go through questions in the previous lecture or AOB~

So here comes the notes of day 1: wk1day1.pdf.

We set the backdrop of integral calculus as the area problem. Along with the tangent problem, these two constitute the two basic problems in Calculus, and one can roughly trace linkage to most ideas in integral and differential calculus respectively starting from here. (See Stewart’s p3-4)

We also explained how a sum of areas of adjacent rectangles $$f(x_i)\triangle x$$ can approximate the area under a curve defined by $$y = f(x)$$. The sum $$\sum_{i=1}^{n} f(x_i) \triangle x$$ is called a Riemann sum and the definite integral is defined to be $$\lim_{n\rightarrow\infty} \sum_{i=1}^{n} f(x_i) \triangle x$$

The whole bunch of symbols in the definition of definite integral means we are approximating the area under a curve by successively increasing the number of subdivisions n, and define the limiting value as the integral value, i.e. the area we’re looking for.

It’s important to note that $$x_i$$ had been chosen, initially, as the right end-point in the i-th interval counting from the left, and this hasn’t to be so, and lefties might prefer the left end points! But it is quite evident from the picture that choosing other points in the interval would not affect the limiting procedure describe above. For all details, please refer to the notes and the textbook, section 5.1 and 5.2.

Our lecture will continue on Tuesday Jan 8th, 9:30-11am (9:15am for early birds who wants to ask questions before the class) and I’ll aim to finish the proof and introduce the first physical interpretation of the area under a curve – distance.

I hope you liked the first lecture, and please let me know what you think by commenting below!

Posted in Lecture
###### 4 comments on “First Lecture”
1. Paniz Banki says:

Thank you for reviewing the stuff we went over in class, and most important what we will be doing in the next class. It makes everything more clear and organize.

• simontse says:

Thank you for your comment, and bringing me to attention that previewing what we’ll do in the next class. Now I find it all the more important to keep orientating you all in the teaching flow.

2. Manolito Pimentel says:

I also just wanted to say thanks for making this blog! It really makes a difference having this kind of up-to-date resource available!

• simontse says:

Thanks, and please then use it more (and Piazza too!), and give me feedback to improve. I’m also learning to teach, and your feedback mean a lot! (Especially if you find the pace going too fast, or something is presented in a confusing way!)