082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

444 Chapter 13 Integration
y
y
y
A(b)
a
A(a)
Figure13.8
Thearea depends onthe limitsaandb.
x
b
x
a
Figure13.9
Thearea betweenx =aandx =bis
A(b) −A(a).
b
x
The expressionA(b) −A(a) isoften writtenas [A(x)] b a . Similarly[x2 +1]
2 3 isthe value
ofx 2 +1atx=3lessthevalueofx 2 +1atx=2.Thus
[x 2 +1] 3 2 =(32 +1)−(2 2 +1)=5
Ingeneral
[f(x)] b a =f(b)−f(a)
Note that since
∫ b
a
ydx =A(b) −A(a)
then, interchanging upper and lower limits,
∫ a
that is,
b
ydx =A(a) −A(b) = −{A(b) −A(a)}
∫ b
ydx=−
∫ a
a b
ydx
Interchanging the limitschanges the sign of the integral.
The evaluation of definite integrals isillustratedinthe following examples.
Example13.7 Evaluate
(a)
∫ 2
1
x 2 +1dx
(b)
∫ 1
∫ 2
Solution (a) LetI stand for x 2 +1dx.
I =
∫ 2
1
1
2
x 2 +1dx
[ ] x
x 2 3 2
+1dx=
3 +x 1
(c)
∫ π
0
sinx dx
Theintegralisnowevaluatedattheupperandlowerlimits.Thedifferencegivesthe
value required.
( ) ( ) 2
3 1
3
I =
3 +2 −
3 +1 = 8 3 +2− 4 3 = 10 3