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

13.3 Definite and indefinite integrals 443
y
y
n
1 2 3 4 n–1
dx
x 1 x 2
dx
x 3 x 4 x 5
x n
x
x 1
x n
x
x
Figure13.6
The area isapproximated by (n −1) rectangles.
Figure13.7
Thearea isextended byadding an extrarectangle.
So,
δA
δx ≈y(x)
Inthe limitas δx → 0,we get
( ) δA
lim = dA
δx→0 δx dx =y(x)
Since differentiation isthe reverse of integration, we can write
∫
A = y(x)dx
To denote the limits of the area being considered we place values on the integral sign.
Thearea under the curve,y(x), betweenx =aandx =bisdenotedas
∫ x=b
x=a
ydx
ormorecompactlyby
∫ b
a
ydx
Theconstantsaandbareknownasthelimitsoftheintegral:lowerandupper,respectively.Sinceanareahasaspecificvalue,suchanintegraliscalledadefiniteintegral.The
area under the curve up to the vertical linex = b isA(b) (see Figure 13.8). Similarly
A(a) is the area up to the vertical linex = a. So the area betweenx = a andx = b is
A(b) −A(a), asshown inFigure 13.9.
Thearea betweenx =aandx =bis given by
Area =
∫ b
a
ydx =A(b) −A(a)
The integral is evaluated at the upper limit, b, and at the lower limit, a, and the
difference between thesegives the required area.