This code is intended to allow students to visualize the logic behind calculating definite integrals using the limit of a Riemann sum. It was written for Political Science 508, a mandatory course for first-semester political science Ph.D. students at Emory University, by Jane Lawrence Sumner.

visualizing-riemann.R

## Definite Integral as the Limit of a Riemann Sum: Visualizing the Logic
## Jane Lawrence Sumner
## TA, POLS 508, Fall 2015


#### INPUTS HERE ####
a <- 1                          ## lower bound
b <- 10                          ## upper bound
n <- 20                         ## number of iterations

fxn <- function(x){
    eq <- x^2                   ## the function to evaluate
    return(eq)
}

#####################

#### WHERE THE MAGIC HAPPENS ####
    
      x <- seq(a-1,b+1,.01)                       ## Set the x-axis to visualize.
      fx <- fxn(x)                                ## Evaluate function for plotting.
      plot(x,fx,type="l")                         ## Plot the function.
      abline(v=c(a,b),lty=3)                      ## Plot the lower and upper bounds.
      abline(h=0)                                 ## Plot the x-axis.
      
      summation <- 0                              ## Start with a sum of zero.
      xlast <- a                                  ## The leftmost bound for boxes.
        for(i in 1:n){
          x <- a+i*(b-a)/n                        ## The formula starts here.
          fx <- fxn(x)                            ## Evaluate function at value.
          segments(x0=xlast,x1=x,y0=fx)        
          segments(y0=0,y1=fx,x0=x)
          polygon(c(xlast,x,x,xlast),c(0,0,fx,fx),col=i,density=45)
          summation <- summation + fx             ## Add this value to previous sum.
          xlast <- x                              ## Set new left bound for box.         
        }
      summation <- (b-a)/n*summation              ## Multiply sum by (b-a)/n
      title(main=paste("The Integral is Approximately ",round(summation,3),sep=""))