Find the area of the region bounded below by the graph of f and above by the x-axis from x = 1 to x = 2.

Question

f(x) = x² − 4x + 3

Answer 1

The area of the region bounded below by the graph of f(x) = x² − 4x + 3 and above by the x-axis from x = 1 to x = 2 is simply this shaded area:

This is a negative area. So put a negative sign before the integral or an absolute value to make it a positive area.

- ∫₁² (x²-4x+3)dx

=-[x³/3 -2x² + 3x]₁²

=-(2³/3 -2•2² +3•2)-[-(1³/3 -2•1² +3•1)]

=-(8/3 -8 + 6)+(1/3 -2 +3)

=-(8/3 - 2) + (1/3 +1)

= -2/3 + 4/3

= 2/3 sq. unit