How do you use Riemann sums to evaluate the area under the curve of #f(x)= 3 - (1/2)x # on the closed interval [2,14], with n=6 rectangles using left endpoints?

1 Answer
Jim H
Nov 7, 2016

Please see the explanation section below.

Explanation:

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

We will approximate.
#int_2^14 (3-1/2x)dx#

Note that #f(x) = 3-1/2x# and #a=2# and #b=14#

#n=6# So #Deltax = (b-a)/n = (14-2)/6 =12/6 = 2#

All endpoints: start with #a# and add #Deltax# successively:

#2# #underbrace(color(white)"XX")_(+2)# #4# #underbrace(color(white)"XX")_(+2)# #6# #underbrace(color(white)"XX")_(+2)# #8# #underbrace(color(white)"XX")_(+2)# #10#
#underbrace(color(white)"XX")_(+2)# #12#
#underbrace(color(white)"XX")_(+2)# #14#

All endpoints: #2#, #4#, #6#, #8#, #10#, #12#, #14#

The subintervals are:

#(2,4)#, #(4, 6)#, #(6,8)#, #(8,10)#, #(10,12)#, #(12,14)#

We have been asked to use the left endpoint of each subinterval.

The left endpoints are:

#2#, #4#, #6#, #8#, #10#, #12#

Now the Riemann sum is the sum of the area of the 6 rectangles. We find the area of each rectangle by

#"height" xx "base" = f("sample point") xx Deltax#

Here we are using left endpoints of subintervals for sample points and #Deltax = 2#. So

#R = f(2)*2+f(4)*2+f(6)*2+ * * * +f(14)*1/2#

#= (f(2)+f(4)+f(6)+ f(8)+f(10)+f(12) +f(14))*2#

Finish the arithmetic to finish.

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