generating and drawing from a trapezoid-rule integration of a bell curve distribution of known total area in R

I would like to draw values from a bell curve shaped distribution. I have a total value for the area under the curve that I would like to be true and would like to designate the total number of trapezoids used.

For example; I would like an approximately-normal distribution with a total area under the curve equal to 200. I want to use 28 trapezoids to make up that total area. so, the 28 trapezoids sum to 200, but the total area of trapezoids 1 and 28 (the ends) are much less than those at positions 14 and 15.

I am modeling a biological process which takes 28 days and costs some amount of resources. Currently I am just dividing the total value (200) by the number of days (28) evenly. But in the interest of modeling the process with a greater amount of realism I'd like the proportion of the total value in the middle days of the process to be greater than the beginning or end, with a smooth transition in between.

This is a problem that I do not know how to solve. I can generate a normal distribution of known mean and sd, but this problem is somewhat different. I want to draw from a sequence of trapezoids which have an associated area value and sum to a parameterized area while following a bell curve shape.

I am aware defining the specifics of the shape of the bell will be necessary, but for the moment, presume any approach that results in a noticeable bell shape with some control over the peaked-ness of the curve would be acceptable.

Considering the bell shape is given by the following function:

(eq1)

the area of it is the following:

So, the area (200) is equal ac*sqrt(2pi) (eq2). Here you have the relationship between a and c.

Considering b=10 (eq3) (which will deslocate the center of the curve to x=10), You still have three equations and four variables.

In order to obtain values to a distribution, you need to define some values to a or c.

library(plotly)
p <- plot_ly()
area <- data.frame()
x = 1:20
ydf <- data.frame("x" = x)
# Escrever excel
for (a in c(30,40,50)) {
  #a <- 100
  b <- 10  
  y = a*exp((-0.5)*((x-b)^2)/((200/(a*sqrt(2*pi)))^2))
  ydf[,as.character(a)] <- y
  p <- p %>% add_trace(x = x, y= y, mode="lines", name = paste("a =",a))
  
  for (i in 2:length(x)){
    area[i-1,as.character(a)] <- ((y[i]+y[i-1])/2)/((x[i]-x[i-1]))
  }
}
p

sum(area$"30")
sum(area$"40")
sum(area$"50")



> sum(area$"30")
[1] 199.8985
> sum(area$"40")
[1] 199.999
> sum(area$"50")
[1] 200

You can see that i needed to set the value of a in order to have a distribution.

You have now the ydf object with the values to y with different a´s, and also the area object, that have the areas of the polygons

This is not an R-specific question. You're just asking for some kind of "reverse" algorithm.

The first problem is that you need to define the limits over which you generate trapezoids. +/- 1e-2? +/- 1e-10 ?
Once you settle that, just use any algorithm to calculate the piece-wise definite integral for each of your 28 slices of the bell curve. Normalize the values thus produced so they add up to whatever total you desire.
I do wonder why you wish to use trapezoids. There are both better (parabolic) and worse (rectangular) approximation methods. What is the accuracy you need?