Lab 15 – Integration and Area
Date: December 6, 2011
Assignment Due Date: December 9, 2011
Goal: In today’s lab we will explore finding approximate areas in R using Riemann sums. We will see
what happens as we make the rectangles smaller and smaller in order to approach exact definite
integrals which we will compute by hand.
Riemann Sums
Integrals can be thought of as taking the area under a curve. We can approximate integrals using
Riemann Sums which involves calculating the area with a finite number of boxes. The figure below
shows a right-hand Riemann sum for the function
from x = 0 to x = 2 with 10 boxes.
The width of each of these boxes is .2 or
. Therefore, the width of each box can be determined by
. The area of each box is simply the width of the box multiplied by the height of
the box which is the function value at the right side of each box for the right-hand Riemann sum. When
we add up the area of all the boxes we get an approximate area under the curve which approximates
the integral
.
Using R to compute Riemann sums
Below is an outline for a script in R to calculate a right-hand Riemann sum as explained before. Fill in the
blanks in order to write a script that produces figures like the one seen above and also computes the
approximate area.
#Define your function
f=
#Define N as number of boxes to use to compute the Riemann sum
N=
#Define the lower limit
lower =
#Define the upper limit
upper =
#Define the width of each box
width =
#Define a vector in order to plot going from the lower limit to the upper limit
x=
#plot the function f(x) as a green line
plot()
#add a line to your plot on the x-axis
lines()
#Initialize area so we can add to it in the for loop.
area =
#begin for loop with the index going from 1 to N for each box
for (){
#define the left x value of the current box
left =
#define the right x value of the current box
right =
#define the height of the current box using the function value at the right side
height =
#add the area of the current box to the total area
area =
#add lines to your plot to represent the current box on your figure
lines(c(left,left,right,right),c(0,height,height,0),col="purple")
}
#Print the area to the console
print(area)
#This code allows you to print the approximate area to the title of your figure.
text = paste("Approximate Area =", as.character(area))
title(main = text)
How would you change this code to calculate the left-hand Riemann sum?
Using your script
Test your code for the function
5.08.
from 0 to 2 with 10 boxes. You should get an answer of
Integrate the function by hand and see what the actual integral is.
See what happens as you increase the number of boxes you use to calculate the Riemann sum.
Compare these results with the answer you computed by hand.
Use your code to approximate the integral of
from 0 to 10. Compare this to the
integral which you compute by hand. Again see what happens as you increase the number of boxes.
What happens if the curve is below the x-axis? To explore this look at the same function from -10 to 10
and also calculate this definite integral by hand.