CHAPTER NAME- INTEGRATION
NCERT class 12 Maths
chapter 1 DEFINITE INTEGRATION GEOMETRICAL INTERPRETATION OF DEFINITE INTEGRAL If f(x) > 0 for all x Î [a, b]; then
ò
b
a
f ( x ) is numerically
x =a
equal to the area bounded by the curve y = f(x), then x-axis
ò
and the straight lines x = a and x = b i.e.
In general
ò
b
a
b
a
x =b S C
f(x) L
f ( x)
+
+
f ( x) dx represents to algebraic sum of the
A
O
-
Q
figures bounded by the curve
M
-
B
D
R
y = f(x), the x-axis and the straight line x = a and x = b. The areas above x-axis are taken place plus sign and the areas below x-axis are taken with minus sign i.e, i.e.
ò
b
a
Note:
f ( x ) dx = area OLA - area AQM – area MRB + area BSCD
ò
b
a
f ( x ) dx = , represents algebraic sum of areas means, that if area of function y = f(x)
is asked between a to b. ÞArea bounded =
ò
b
a
b
f(x) dx andnot beenrepresented by ò f(x) dx a
e.g., If some one asks the area of y = x3 between -1 to 1. Then y = x3 could be plotted as; \ Area =
ò
0
-1
1
-x 3 dx + ò x 3 dx = 0
or, using above definition Area =
1
-1
1 2
O -1
ò
1
-1
1
x dx = 2ò x 3 dx 3
0
1
é x4 ù 1 =2ê ú = ë 4 û0 2 But if, we integrate x3 between -1 to 1.
y = x3
1