How do you find an approximation for the definite integrals #int 1/x# by calculating the Riemann sum with 4 subdivisions using the right endpoints from 1 to 4?

1 Answer
Oct 25, 2015

Assuming equal subdivisions, see the explanation section below.

Explanation:

I will use what I think is the usual notation throughout this solution.

#int_1^4 1/x dx#

Note that #f(x) = 1/x# and #a=1# and #b=4#

#n=4# So #Deltax = (b-a)/n = (4-1)/4 =3/4#

To find all of the endpoints: start with #a# and add #Deltax# successively:

#1# #underbrace(color(white)"XX")_(+3/4)# #7/4# #underbrace(color(white)"XX")_(+3/4)# #10/4# #underbrace(color(white)"XX")_(+3/4)# #13/4# #underbrace(color(white)"XX")_(+3/4)# #16/4#

Right endpoints: #7/4#, #5/2#, #13/4#, #4#

Now the Riemann sum is the sum of the area of the 4 rectangles. We find the area of each rectangle by
#"height" xx "base" = f("endpoint") xx Deltax#

So

#R = f(7/4)*1+f(5/2)*1+f(13/4)*1+f(4)*1#

#= 4/7+2/5+4/13+1/4#

To finish, do the arithmetic.