Page 46 - Math SL HB Sem 2

The area of the lower rectangle is f(x) x 6a and that ofthe upper rectangle is f(x + 6x) x 0x.
b-&
Then the sum of the areas of the lower rectangles for a < x < b is S,_:
Zf <lA .

and the sum of the areas of the upper rectangles for a < x < b is S, = lf @ @& .
+
:
If A unit2 is the area under the curve y (x) over the interval a,b ] , we have that
I
b& b
Zf<oa <Aalfe+e)e

As the number of strips increase, 1 ---+co, and therefore 6x -' 0, then the areqA sq. unitrwill
approaches a common limit, i.e. Srfrom below, and Su from above. We write this result as:

A= limt/(r)&

b
lroa,

Area betwee 'n the Curve and the x- axis
Def: If fis continuous throughout a, b ], then the area ofthe region between the curve
I
:
-
y (x) and the x axis from a to b is given by the formula
v
f r1*ya,
rf(x)

x
a b

]
If fis negative over the interval a,b (i.e.f(x)<0fora 3 x S b ), then the integral
I
A
I (x) ax is a negative number. We therefore need to write the are4 A, as

v
a b
f q,.l*

Or o:lfr(x)dxl
(the absolute value ofthe integral) v:f(x)