# age by age-bins
age <- c(5, 10, 20, 50, 100, 200)

# fractional area by age-bin
farea <- rep(1/6, 6)

# some function of similar form as in your paper (negative coefficienty, log of age)
calc_bp <- function(age){10 - 1 * log(age)}

# calculate BP by age bin and plot
bp <- calc_bp(age)
df <- tibble(age, bp)
df %>% 
  ggplot(aes(age, bp)) + 
  geom_point()

Area-weighted mean(f(age)):

Calculate BP by age bin and take area-weighted mean (multiply by farea and take sum).

sum(calc_bp(age) * farea)
## [1] 6.546122

f( area-weighted mean(age)):

Calulate a single BP as a function of area-weighted mean age.

calc_bp(sum(age * farea))
## [1] 5.838516

The latter is smaller than the former. In other words, BP and N-uptake area larger when calculating BP by age bin and then take area-weighted mean .

One could also prove this mathematically, given that the second derivative of BP vs. age is positive.