How to calculate the cumulative area of the overlapped normal distribution curve in R ?

General
6 
mean_sim <- 5
up_lmt <- (max(1:10) * 2 * mean_sim)

u <- seq(from = 0,
         to = up_lmt,
         length.out = 1e+3)
v <- sapply(X = 1:10,
            FUN = function(i) dnorm(x = u,
                                    mean = (i * mean_sim),
                                    sd = i))

matplot(x = u,
        y = v,
        type = "l",
        lty = 1,
        col = rainbow(n = 10),
        xlab = "Time",
        ylab = "Probabilities",
        main = "Cumulative Area")
legend(x = "topright",
       title = "Signal Number",
       legend = 1:10,
       col = rainbow(n = 10),
       bty = "n",
       lty = 1)

shade_and_find_area <- function(start_pt, end_pt, colour = "grey")
{
  idx <- ((u >= start_pt) & (u <= end_pt))
  u_obj <- u[idx]
  v_obj <- apply(X = v[idx, ],
                 MARGIN = 1,
                 FUN = max)
  
  polygon(x = c(start_pt, u_obj, end_pt),
          y = c(0, v_obj, 0),
          border = NA,
          col = adjustcolor(col = colour,
                            alpha.f = 0.5))
  
  DescTools::AUC(x = c(start_pt, u_obj, end_pt),
                 y = c(0, v_obj, 0))
}

shade_and_find_area(start_pt = 0,
                    end_pt = 10,
                    colour = "blue")
#> [1] 1.401842

shade_and_find_area(start_pt = 20,
                    end_pt = 30,
                    colour = "red")
#> [1] 0.7824488

shade_and_find_area(start_pt = 40,
                    end_pt = 50,
                    colour = "green")
 #> [1] 0.4452164

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