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THE DEFINITE INTEGRAL

Calculus consists of two main parts, diferential and integral calculus. Diferential calculus is based upon the

derivative. In this text we define the concept that is the basis for integral calculus: the definite integral. The

bridge between these two different parts, that we shall discuss later in math class, is the  Fundamental

Theorem of Calculus. The integral of a  function is one of the most powerful tools in mathematics and the

applied sciences. The definition of the definite integral is closely related to the areas of certain regions in a

coordinate plane. We can easily calculate the area if the region is bounded by lines. But if this is not the case,

we must introduce a limiting process and then use methods of calculus.

In particular, let us consider the problem to find the area of a given region  A in the plane, bounded by the

vertical line x=a, and x=b, by the x axis , bounded above by the graph of a positive function  f(x) definite on a

closed interval [a, b]. A region of this type is illustrated in  Figure 1 , where the curve “ y=f(x)” is not

necessarily a straight line.

Figure 1: Our objetive is to define the area of  A.

The answer to this problem came through a very nice idea. Indeed, let us split the region  A into small

subregions which we can approximate by rectangles or other simple geometrical figures (whose areas we

know how to compute). This is how it goes: split the interval [a, b] into n subintervals , preferably with the

same width ∆x.

 x0=a x

1 x

2⋯ xn=b with  xi− x i−1

=  x =b−a

n  , for i= 1, 2, 3,  , n.

Note that: x

0= a , x

1= x

0   x , x

2= x

0 2 x ,  ,

 xi= x0 i x ,  , x n=  x

0 n x= a n b−a

n = b.

 

 Mat. Osman Villanueva Garcí a Page 1 de 3

ba

“X” axis

“Y” axis

y=f(x)

 A

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If we want to approximate the area of the region  A, by rectangles, then we only have to worry about the

upper boundary of each rectangle (since each subinterval represents the basis of each rectangle). The easiest

way to choose a height for our rectangles is to choose the value of the function at the left (see  Figure 2a ) end

points, or at the right ( Figure 2b ) end points of the small intervals.

 Figure 2a: Aproximation of  A using rectangles with height at the left value of each “∆  x” .

Note that: Sn= R1 R

2⋯ Rn implies: Sn= x [ f  x

0 f  x

1 f  x

2⋯ f  xn−1

]

using summation notation: Sn= x [∑i =0

n−1

 f  xi ] =  x [∑i=1

n

 f  xi−1 ] .

 Figure 2b: Aproximation of  A using rectangles with height at the right value of each “∆  x” .

Note that: Sn= R1 R

2⋯ Rn implies: Sn= x [ f  x

1 f  x

2 f  x

3⋯ f  xn]

using summation notation:Sn= x [∑

i=1

n

 f   xi] .

 Mat. Osman Villanueva Garcí a Page 2 de 3

(x2 , f(x

2))

(xn , f(x

n))

a=x0

“X” axis

“Y” axis

y=f(x)

Sn

(x1 , f(x

1))

(xn 1

  , f(xn 1

))(x3 , f(x

3))

 x1

 x2

 x3 .... b=x

n x

n 1 x

n 2

 ....

 ....∆ x ∆ x ∆ x ∆ x ∆ x

 R1

 R2

 R3

 Rn 11

 Rn

(x2  , f(x

2))

(xn 2

  , f(xn 2

))

a=x0

“X” axis

“Y” axis

y=f(x)

Sn

(x0  , f(x0))

(xn 1

  , f(xn 1

))

(x1  , f(x

1))

 x1

 x2

 x3 .... b=x

n x

n 1 x

n 2

 ....

∆ x∆ x ∆ x ∆ x ∆ x ....

 R1

 R2

 R3

 Rn

 Rn 11

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Referring to both Figures (2a and 2b) it appears that: Sn A Sn . If the number of rectangles “n”

is very large or, equivalently, if “∆ x” is very small, then the sum of the rectangular areas should be close to

the area of the region A that we want to define.

As you can see, when “∆ x” get closer and closer to “ zero”, then “n” increases without bound and

consequently we will observe that Sn and Sn approaches to the value of  A. This is the concept of limit

once again! In other words, we have:

lim Sn = A = lim Sn ⇔ lim x 0

[∑i=0

n−1

 f  xi  x ] =  A = lim

 x0[∑

i=1

n

 f   xi  x ]

provided the limit exists.

The preceding limit is one of the fundamental concepts of calculus. It is called the definite integral of the

 function “f” from “a” to “b”.

Definition 1 (The definite integral of a funtion): Let f be a function that is defined on an closed interval [a,

b]. The definite integral of the function “f” from “a” to “b” , denoted by 

∫a

b

 f  x dx , is:

∫a

b

 f  x dx= limn ∞ [∑

i=1

n

 f  x i−1  x ]= lim

n ∞ [∑i=1

n

 f  x i  x ]= lim x 0 [∑i

 f  xi  x ]provided the limit exists.

If the definite integral of “ f ” from “a” to “b” exists, then f is integrable on [a, b], and we say that the integral

∫a

b

 f  x dx exists. The symbol ∫   is an integral sign (the first letter of the word sum). The number a

and b are the limits of integration. The expression  f(x) is calling the integrand. The diferential symbol dx

must be associated with the increment ∆ x.

 Mat. Osman Villanueva Garcí a Page 3 de 3

ba

“X” axis

“Y” axis

y=f(x)

 A=∫ f(x)dxa

b