How do you estimate the area under the graph of #f(x)=25-x^2# from #x=0# to #x=5# using five approximating rectangles and right endpoints?

1 Answer
Mar 14, 2015

We are approximating an area from #a# to #b# with #a=0# and #b=5#, #n=5#, right endpoints and #f(x)=25-x^2#
(For comparison, we'll do the same problem, but use left endpoints after we finish this.)

We need # Delta x=(b-a)/n#

#Deltax# is both the base of each rectangle and the distance between the endpoints.

For this problems #Deltax=(5-0)/5=1#.

Now, find the endpoints. (All of them to start with.)

The left-most endpoint is #a#, which, in this problem, is #0#. Start adding #Deltax# until we get to the end of the Intevral we are interested in.

Endpoints: #a=0#,
#a+Deltax=0+1=1#,
the next endpoint is the previous endpoint plus #Deltax#, #1+Delta x= 1+1=2#,
then #2+1=3#, and so on 4,# and #5#.

The endpoints are: #0,1,2,3,4,5#.
The right endpoints are #1,2,3,4,5#

The heights at these endpoints are:
#f(1)=24#
#f(2)=21#
#f(3)=16#,
#f(4)=9# and
#f(5)=0#

The areas of the rectagles are #Deltax# times the heights.

#1*24=24#,
#1*21=21#,
#1*16=16# and so on.

The area can be approximated by adding the areas of the five rectangles:
#(1*24)+(1*21)+(1*16)+(1*9)+(1*0) =70#

We did not use the graph of the function, but here it is, if you want to look at it.

graph{25-x^2 [-4.72, 46.6, -1.03, 24.65]}

By way of comparison: Using the LEFT endpoints and #5# rectangles would have given us:
The LEFT endpoints are #0, 1,2,3,4,#

The heights at these left endpoints are:
#f(0)=25#
#f(1)=24#,
#f(2)=21#,
#f(3)=16#, and
#f(4)=9#

The areas of the rectagles are #Deltax# times the heights.

#1*25=25#,
#1*24=24#,
#1*21=21# and so on.

The area can be approximated by adding the areas of the five rectangles:
#(1*25)+(1*24)+(1*21)+(1*16)+(1*9)=95#.