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.

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|

3.6 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

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

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

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