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?

Notice that if you take 1- each step height, you will get the "un-step" height of 1/(2^(k+1)) as length and since the width of the "un-steps" is equal to the width of the steps, which is (1-1/(2^k))-(1-1/(2^(k+1))) or 1/(2^(k+1), the area is just the sum of 1/(2^2k) so the answer is 1-(1/4 + 1/16 + 1/64 ...) or 1-((1/4)/(3/4)) or 1-(1/3) or 2/3