Mathematics -Class XII

Unit III -Calculus

Chapter VII -Integrals

Topic -Definite Integrals

Sub Topic -Definite Integral as

the Limit of a Sum –

Definition and proof

Outline:

• Definition
• Integral as the Limit of a sum –proof
• Example with solutions
• Assignment I
• Assignment II

DEFINITE INTEGRAL

AS THE LIMIT OF SUM

(PART-I)

DEFINITION

Let f be a continuous function defined on the closed interval [a, b],

• Let a = x₀ < x₁ < x₂ , … < x_{n – 1} < x_n = b .

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• Divide [a,b] into n parts with width h=(b-a)/n.

As n ∞

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is called definite integral of f from a to b.

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The definite integral

is the area bounded by the curve

y= f(x)., the ordinates x=a , x=b and the x-axis

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To evaluate this area , consider the region PRSQP between the curve, x-axis and the ordinates at x=a and x=b.

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Y

0

X

Q

P

R

S

x=a

x=b

Divide the interval [a,b] into n equal subintervals denoted by

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Y

0

X

_Xn=b

_X0=a

_X1

_X2

_X3

_xr-1

_xr

Q

P

R

S

A

B

L

D

M

C

[x₀,x₁],[x₁,x₂],….[x_n-1,x_n] , where x₀=a, x₁=a+h, x₂=a+2h…, x_r=a+rh , r=1,2,3,…n

and x_n=b=a+nh, or h= (b-a)/n. Also as

The region PRSQP under consideration is the sum of n sub regions where each sub region is defined on the subintervals [x_r-1,x_r], r=1,2,3,…n. hence from fig.

Area of the rectangle (ABLC)<area of the region (ABDCA)< area of rectangle(ABDM )

Y

0

X

A

B

L

C

M

D

_xr-1

_xr

As x_r-x_r-1 0,i.e ,h 0 the three areas are nearly equal .

Here s_n & S_n are sum of the areas of all lower rectangles and higher rectangles over subintervals [x_r-1,x_r], r=1,2,3,…n.

Therefore s_n<area of the region PRSPQ<S_n

or

As , all the three areas are equal

Applications

• Evaluate: as limit of a sum?

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Ans.

SOLUTION BY SECOND FORMULA

Here a=0, b=2 , h=2/n

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Solution contd…