DinS          Written on 2018/4/25

 The concepts of differentiation and integration are related

1. Curved area

What’s area? This seems plain enough.

For the above rectangle, its width is 2cm and height 1cm. We say for sure that its area is 2 * 1 = 2cm². No problem, but what do we mean that the area of the rectangle is 2cm²? We can cut the rectangle into pieces and arrange them so that the pieces together can cover an area of 2cm².

But when we come to curved areas, it doesn’t work. The area of the circle above is π. We can’t, however, rearrange it to a rectangle with height of 1cm and width of π cm.

We need to redefine area with curves. The tool we have is limit.

When calculating curved areas, we make up tiny rectangles to fill the area. If the rectangles are small enough, by small enough that is so say the width to close to 0, they can cover the whole area. We know how to calculate area for rectangle for sure. If we sum all the area of rectangles up, we get the curved area. This method is called Riemann Sum.

This graph shows the idea of Riemann Sum. We can calculate each width and corresponding height f(x). As a result we can calculate all white rectangles. If we sum up all rectangles, we’re approximating the curved area. The more the width is close to zero, the more rectangles we have, and the more close we get to the real area.

Let’s generalize the steps for Riemann Sum. First partition the interval [a, b] so that we have

a = x₀ < x₁ < x₂ < x₃ < …… < x_n = b

Next choose sample points . If  that’s left Riemann Sum. If  that’s right Riemann Sum.

Then we can write the formula:

Really this is not complicated. It’s just a translation from verbal description to math notation.  is the height for each rectangle, and  is the width for each rectangle. If we multiply these two we get the area of a single rectangle. And if we sum all rectangles we get the area.

Notice it’s not the equal sign because if the partition is wide we will lose some areas.

2. Integral

Math has a tendency to get rid of natural language and use notations instead. The word “area” above is not math enough. Thus we use a fancy notation to represent curved area.

This is called integral. What’s that mean?

You can try to think of it this way. The dx is the width for each rectangle and f(x) the corresponding height. f(x)dx then is the area of each rectangle. The symbol ∫ means sum up all the rectangles from boundary a to b.

The explanation above may be a bit misleading. If it helps you to understand the symbols, however, that should be good enough.

Armed with the integral symbol, we can now define curved area.

Nothing special. We just add the limit to Riemann Sum and replace the word “area” with the symbol of integral.

Notice that to say  means that we’re stating the limit is close to I.

How to calculate integral? See an example in action.

First we partition the interval [0, 1] to n parts and we’ll use right Riemann Sum. Thus we have:

Thus we have:

We can pull out the n³ since it has nothing to do with i. What about i² sum? Luckily we know a formula to calculate this.

We can use this formula directly. Now all works done, time for the integral.

We did it! Actually this is quite hard. We’re just doing very simple functions like x². If the function gets complicated, things could go pretty ugly.

I should remind you of what I said in the limit section. Limit serves as the founding block for integral. Now we’re calculating integral using limit. It is to say we’re doing it with bare hands. In real problem we have more decent methods to do this. But before jumping in, let’s first understand integral a bit more.

3. Understanding integrals

Here are a few rules for integral.

Once you draw a graph you’ll see it’s reasonable.

We can pull the constant out. It can also be explained via drawing a graph.

Here’s an important concept called accumulation function. See a graph below.

Accumulation function A(x) is defined as

Given a fixed point a, accumulation function A(x) measures the area from a to x. It’s call accumulation because it describes the accumulated change with regard of areas from a to x. And notice that integral actually measures signed area. f(t) can be negative.

f positive -> A(x) increasing -> A’(x) > 0

f negative -> A(x) decreasing -> A’(x) < 0

So you see there’re some relationships between derivatives and integrals. Let’s explore that a little more.

4. Fundamental Theorem of Calculus

Newton and Leibniz are credited for discovering calculus because they found this theorem.

Statement:

Suppose f:[a, b]->R is continuous. Let F be the accumulation function, given by

Then F is continuous on [a, b], differentiable on (a, b) and F’(x) = f(x).

Brief proof:

We have  . Now we add a small increase of h to x. As a result the increased area will be F(x+h) – F(x), the shadows region above. Now since h is small enough we take that region as a rectangle. The width is h and height is f(x). Thus we have:

h is small enough means h is close to 0. We can write as this:

This means  is an anti-derivative of f! Amazing!

In real problem we are more concerned about 

Here’s a transformation that will help.

Suppose G is an anti-derivative of f, i.e. G’(x) = f(x). Besides we have F’(x) = f(x).

Thus we have

F(x) = G(x) + c  ①

Because the difference between anti-derivatives lies in constant term.

From the definition of accumulation function we know 

We replace x with a in ① and get F(a) = G(a) + c = 0. Thus c = -G(a).

Now we replace c with –G(a) in ① and get

From here we can draw another conclusion.

Statement:

Suppose f:[a, b]->R is continuous, and F is an anti-derivative of f. Then

This is a more easy-to-apply version of Fundamental Theorem of Calculus.

Let’s try out this amazing tool.

We’ve solved it using limit and Riemann Sum. Let’s do it with F.T.C.

First thing is to find an anti-derivative of x². That’s easy .

Now applying F.T.C. we have

Pretty cool right?

We can make it even more succinct.

Using this fancy notation we can get rid of constant term and don’t need to explicitly express F(x).