From [0, 1/2), the step is 1/2 high.
From [1/2, 3/4), the step is 3/4 high
From [3/4, 7/8), the step is 7/8 high
From [ 1 - 1/(2^k), 1 - 1/(2^(k+1)) ) the step is 1 - 1/(2^(k+1)) high

What is the exact area under the step function?

The width of each vertical strip making up the area is 1/2^(k+1), while the height is 1-1/2^(k+1).  The area of each strip is therefore 1/2^(k+1) - 1/4^(k+1), with k going from zero to infinity.

Let a be the sum of all the positive terms and b be the sum of the absolute values of the negative terms. The answer will be a - b.

a = 1/2 + 1/4 + 1/8 + 1/16 + ...

2a = 1 + a

a = 1

and

b = 1/4 + 1/16 + 1/64 + 1/256 + ...

4b = 1 + b

3b = 1

b = 1/3

Total area  =  a - b = 1 - 1/3  =  2/3.