1. 3.6 Area enclosed by the x-axis
Areas above the x-axis give a positive
Definite Integral
Areas below the x-axis give a negative
Definite Integral
We always think of area as positive.
Therefore Area = |ab f(x).dx|
Always sketch the area to be found.

2. 3.6 Area enclosed by the x-axis
a b c
Some functions have area above and below the x-axis.
The Definite Integral |ac f(x).dx| < Area.
Therefore Area = |ab f(x).dx| + |bc f(x).dx|

3. 3.6 Area enclosed by the x-axis
Example 1:
Find the area between y = x2 – 4 and x-axis.
(x2 -4).dx  -2 2
Area = x3 3 [ ]
- 4x -2 2 = -2 2
x2 – 4 = 0 23 3 [ ]
- 4x2 =
-23 3 [ ]
- 4x(-2) -
x2 = 4
x = ±2
≈ |-10.7|
≈ 10.7

4. 3.6 Area enclosed by the x-axis
Example 2:
Find the area between y = x3 and x-axis from -1 to 1.
x3.dx  -1
x3.dx  1 +
Area = -1 1 x4 4 [ ] -1 = x4 4 [ ] 1 + -1 4 = + 1 4 = 1 2

5. 3.6 Area enclosed by the x-axis
For Odd Functions  -a a
f(x).dx = 0
For Even Functions  -a a
f(x).dx < 0 a -a
So for Odd & Even Functions  -a a
f(x).dx =
2 f(x).dx

6. 3.6 Area enclosed by the x-axis
Example 2:
Find the area between y = x3 and x-axis from -1 to 1.
x3.dx  1 2x
Area = -1 1 [ ] 1 x4 4
= 2x 1 4 = 2x = 1 2