How do you use Riemann sums to evaluate the area under the curve of #f(x) = x^2 + 3x# on the closed interval [0,8], with n=4 rectangles using midpoint?

1 Answer
Jan 22, 2016

Please see the explanation section below.

Explanation:

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

Note that #f(x) = x^2+3x# and #a=0# and #b=8#

#n=4# So #Deltax = (b-a)/n = (8-0)/4 =2#

The endpoints: star with #a# and add #Deltax# successively:

#0# #underbrace(color(white)"XX")_(+2)# #2# #underbrace(color(white)"XX")_(+2)# #4# #underbrace(color(white)"XX")_(+2)# #6# #underbrace(color(white)"XX")_(+2)# #8#

The midpoints may be found by averaging the endpoints.
They are: #1#, #3#, #5#, #7#

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("midpoint") xx Deltax#

So

#R = f(1)*2+f(3)*2+f(5)*2+f(7)*2#

#= (f(1)+f(3)+f(5)+f(7))*2#

The arithmetic is left to the student.