Is the probability of a continuous variable obtained via integrating over an interval of the probability density curve *cumulative* probability? set.seed(42)
a <- rnorm(200) # generate random data
hist(a, probability = T) # histogram
d <- density(a) # calculate density
polygon(d, density = 10, col = "deeppink") # plot density
abline(v=0, col="blue") # draw boundaries
abline(v=0.1, col="blue")

If I want to know the probability of the variable between 2 points (since it is considered continuous), e.g. 0 and 0.1, the way to calculate it is:
integrate(approxfun(d), lower = 0, upper = 0.1)

Since the probability density function is the derivative of the cumulative distribution function, when I apply integrate(approxfun(d), lower = 0, upper = 0.1), I am reversing the derivation. Does this mean that the result I get is cumulative  probability?

The cumulative density function (cdf, $F(x)$) and the probability density function (pdf, $f(x)$) are related by the equation:
$F(x) = P(X \leq x) = \int_{-\infty}^{x} f(x)dx$
By the fundamental theorem of calculus, we have that $\frac{\partial}{\partial x}F(x) = f(x)$
(We can replace $-\infty$ with the lower limit of the distribution if that distribution doesn't have support extending to $-\infty$.)
For continuous distributions, we can show that $P(X = a) = 0$ for all points $a$. So we can be a little flexible with our uses of $<$ vs $\leq$.
The probability that $X$ takes a value in a range is defined as
$P(a < X < b) = \int_{a}^{b}f(x)dx$
Which is also equal to
$P(a < X \leq b) = F(b) - F(a)$
So when we take an integral $\int_{a}^{b}f(x)dx$, we are not getting a cumulative probability, except in the case that $a = -\infty$ (or the lower limit of the distribution).
