Finding Area Under Curve

Question

I would like to find the area under the curve that is produced by running the following code:

L <- 11 #increments

#time vector
t0 <- 0
t1 <- 1
t <- seq(from = t0, to = t1, by = (t1-t0)/(L-1))

#mass flow rate
m <- c(0.00000, 85.44207, 247.48737, 519.72348, 987.59247, 1679.37861, 2629.25871, 
   3535.53391, 4378.75874, 5020.45815, 5288.56947)

plot(t, m, type = "l", col = "red", lwd = 5)

Answer 1

Probably you can try interpolation first and then integrate, e.g.,

> fl <- \(x) approxfun(t, m)(x) # linear interpolation

> integrate(fl, 0, 1)$value
[1] 2172.785

or

> fs <- \(x) splinefun(t, m)(x) # spline interpolation

> integrate(fs, 0, 1)$value
[1] 2173.054

If your curve consists of linear segments, you can use the following two simple approaches as well

> # if step size is NOT uniform
> sum(rowSums(embed(m, 2)) * diff(t)) / 2
[1] 2172.792

> # if step size if uniform
> (sum(m) - mean(range(m))) * (t1 - t0) / (L - 1)
[1] 2172.792

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To visualize the interpolated curves, you can refer to the following example

plot(t, m, type = "p", lwd = 3)
curve(fl, 0, 1, add = TRUE, col = "blue", lwd = 2)
curve(fs, 0, 1, add = TRUE, col = "red", lwd = 2)
legend(
    x = "topleft",
    legend = c("data", "approxfun", "splinefun"),
    lwd = 2,
    col = c("black", "blue", "red"),
    text.font = 2
)
grid(10)