The Manga Guide to Calculus

Iustrating the Fundamental Theorem of Calculus 87

Then, if we make this

division infinitely fine,

we wi get the exact

amount of alcohol,

won’t we?

We, that’s

true, but it’s

not realistic.

You’d have to a up

an infinite number of

infinitely fine portions.

Lk at this

expreion. Does

it remind you of

something?

Ah!

It lks like an

imitating linear

function!

I...

I s.

p x x x

3 4 3

( )

× −

( )

88 Chapter 3 Let’s Integrate a Function!

Step 4—Review of the Imitating Linear Function

When the derivative of f(x) is given by f

′ 

(x), we had f(x) ≈ f

′ 

(a) (x − a) + f(a)

near x = a.

Transposing f(a), we get

u

f x f a f a x a

( )

−

( )

≈

′

( )

−

( )

or (Difference in f ) ≈ (Derivative of f ) × (Difference in x)

If we assume that the interval between two consecutive values of x

0

, x

1

,

x

2

, x

3

, ..., x

6

is small enough, x

1

is close to x

0

, x

2

is close to x

1

, and so on.

Now, let’s introduce a new function, q(x), whose derivative is p(x). This

means q

′ 

(x) = p(x).

Using u for this q(x), we get

(Difference in q) ≈ (Derivative of q) × (Difference in x)

q x q x p x x x

1 0 0 1 0

( )

−

( )

≈

( )

−

( )

q x q x p x x x

2 1 1 2 1

( )

−

( )

≈

( )

−

( )

The sum of the right sides of these expressions is the same as the sum

of the left sides.

Some terms in the expressions for the sum cancel each other out.

q x q x p x x x

q x q x p x x x

q x q x

1 0 0 1 0

2 1 1 2 1

3 2

( )

−

( )

≈

( )

−

( )

( )

−

( )

≈

( )

−

( )

( )

−

( ))

≈

( )

−

( )

( )

−

( )

≈

( )

−

( )

( )

−

( )

≈

( )

p x x x

q x q x p x x x

q x q x p x x

2 3 2

4 3 3 4 3

5 4 4 5

−−

( )

+

( )

−

( )

≈

( )

−

( )

x

q x q x p x x x

4

6 5 5 6 5

q x q x

6 0

( )

−

( )

≈

The sum

Substituting x

6

= 9 and x

0

= 0, we get

The approximate amount of alcohol = the sum × 20

q x q x

6 0

20

( )

−

( )

{ }

×

q q9 0 20

( )

−

( )

{ }

×

So we nd to find

function q(x) that

satisfies

q

′ 

(x) = p(x).

Iustrating the Fundamental Theorem of Calculus 89

We have just

obtained the

foowing

relationship of

expreions

shown in the

diagram.

But if we increase

the number of

points

x

0

, x

1

, x

2

, x

3

,

and so on, until it

becomes infinite,

we can say that

relationship u

changes from

“aroximation”

to“equality.”

But, since the sum

of the expreions

have bn imitating

the constant value

q(9) − q(0) ,

we get the

relationship

shownhere.

*

Step 5—Approximation



Exact Value

* We wi obtain this relationship

more rigorously on page 94.





The approximate amount of alcohol

(÷ 20) given by the stepwise function:

(Constant)

The exact amount

of alcohol (÷ 20)

≈

≈

The exact amount

of alcohol (÷ 20)

=

=

=

The sum of

for an infinite number of x

i

p x x x

i i i

( )

−

( )

+1

q q9 0

( )

−

( )

p x x x p x x x

0 1 0 1 2 1

( )

−

( )

+

( )

−

( )

+ ...

q q9 0

( )

−

( )