I understand this question is about two years old, but it was recently poked by Community and so came to my attention. I'd solve this problem using the relationship between a survival function of a non-negative random variable and its expectation. Let $X$ be a non-negative random variable (here $X$ is total sales). Then $EX = \int_0^\infty (1-F(x))\ dx$, where $F(x) = P(X \leq x)$ denotes the cumulative distribution function (CDF) of $X$. From the data you have, we can see that $P(100 \leq X \leq 200) = 1$ (implying $X$ is non-negative). So, we have
$$
\begin{aligned}
EX &= \int_{0}^{\infty} (1-F(x)) \ dx \\
&= \int_{0}^{200} (1-F(x)) \ dx \qquad \text{(since } 1 - F(x) = P(X > x) = 0 \text{ for } x \geq 200) \\
&= (200 - 0) - \int_{100}^{200} F(x) \ dx \\
&= 200 - \left[\int_{100}^{105} F(x) \ dx + \int_{105}^{110} F(x) \ dx + \dots + \int_{171}^{200} F(x) \ dx \right]
\end{aligned}
$$
Since we don't know the CDF completely, all we can do is some approximation. A simple (albeit crude) approximation is given by the composite trapezoidal method. Here we approximate each of the above sub-integrals using the Trapezoidal rule, which says that the definite integral
$$
\int_{a}^{b} F(x) \, dx \approx (b-a) \cdot \frac{F(a)+F(b)}{2}.
$$
So, e.g., $\int_{100}^{105} F(x) \ dx \approx (105 - 100) [F(100) + F(105)]/2 = 5 * (0 + 0.05)/2 = 0.125$. An R implementation of this approximation scheme is as follows:
x <- c(100, 105, 110, 116, 122, 128, 134,
141, 148, 155, 163, 171, 200)
Fx <- c(0, 0.05, 0.10, 0.15, 0.20, 0.25, 0.35,
0.45, 0.55, 0.65, 0.75, 0.90, 1)
n <- length(Fx)
EX.approx <- 200 - sum(diff(x) * (Fx[-n] + Fx[-1])/2)
EX.approx
# [1] 144.3
Another approximation will be obtained by first linearly interpolating $F$ in $[100, 200]$ based on the data we have, and then doing a numerical integration over that interpolated function. Here is an R implementation based on this interpolation approach:
Fx_smooth <- approxfun(x = x, y = Fx, method = "linear")
EX.approx2 <- 200 - integrate(Fx_smooth,
lower = 100,
upper = 200)$val
EX.approx2
# [1] 144.2997
Edits: Grammar.
