3.5 Integration as the Summation of Areas

3.5 Integration as the Summation of Areas

(A) Area of the region between a Curve and the x-axis.



Area of the shaded region;  A= a b y dx


(B) Area of the region between a curve and the y-axis.


Area of the shaded region;  A= a b x dy


(C) Area of the region between a curve and a straight line.


Area of the shaded region;  A= a b f(x) dx a b g(x) dx


Example 1
Find the area of the shaded region.


Solution:
Area of the shaded region = a b y dx = 0 4 ( 6x x 2 )dx = [ 6 x 2 2 x 3 3 ] 0 4 =[ 3 (4) 2 ( 4 ) 3 3 ]0 =26 2 3  unit 2


Example 2
Find the area of the shaded region.


Solution:
y = x -----(1)
x = 8yy2 -----(2)
Substitute (1) into (2),
y = 8yy2
y2 – 7y = 0
y (y – 7) = 0
y = 0 or 7
From (1), x = 0 or 7
Therefore the intersection points of the curve and the straight line is (0, 0) and (7, 7).

Intersection point of the curve and y-axis is,
x = 8yy2
At y-axis, x = 0
0 = 8yy2
y (y – 8) = 0
y = 0, 8

Area of shaded region = (A1) Area of triangle + (A2) Area under the curve from y = 7 to y = 8.
= 1 2 ×base×height +  7 8 x dy = 1 2 ×( 7 )( 7 )+ 7 8 ( 8y y 2 ) dy = 49 2 + [ 8 y 2 2 y 3 3 ] 7 8 =24 1 2 +[ 4 (8) 2 ( 8 ) 3 3 ][ 4 (7) 2 ( 7 ) 3 3 ] =24 1 2 +85 1 3 81 2 3 =28 1 6  unit 2