Finding area between functions

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let s(eps) be the face of a figure bounded by y = e^x and y=x+1+eps. We assume that eps is a small positive number. At eps->0 we have S(eps) = S1+eps^a*S2+... a>0. Find S1 S2 and a.
 
To find the value of S1, S2, and a as ε approaches 0 for the figure bounded by y=e^x and y=x+1+ε, we need to analyze the area of the bounded region.

As ε tends to 0, the curve y=x+1+ε approaches the line y=x+1. So, the bounded region is effectively between the curves y=e^x and y=x+1.

To find S1, we need to find the area between the curves y=ex and y=x+1. This can be calculated by integrating the difference of the functions over the appropriate interval. S1 represents this limiting area as ε approaches 0.

For S2, we'll look at the next term in the expansion. It involves the area of a region that scales with εa, where a>0. This suggests that as ε tends to 0, the contribution of this term becomes increasingly significant relative to S1.

To find a, we analyze how quickly the area enclosed by the curves changes as ε approaches 0. This is encapsulated in the power a.


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