How do you estimate the area under the graph of #f(x) = sqrt x# from #x=0# to #x=4# using four approximating rectangles and right endpoints?

1 Answer
Mar 24, 2015

For this problem: #f(x)=sqrtx#

#a=0# and #b=4#

the number of rectangles #=n=4#

#Delta x =# the length of each subinterval = the length of each base

#Delta x = (b - a)/n = (4 - 0)/4=1#

To find all of the endpoints of subintervals, start at #a# and successively add #Delta x# until you reach #b#

All endpoints: #0, 1, 2, 3, 4#.

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

The heights at the right endpoints are:

#f(1)=sqrt1=1#
#f(2)=sqrt2#
#f(3)=sqrt3#
#f(4)=sqrt4=2#

Call the areas of the rectangles #R_1, R_2# etc: Each has area #"base" xx "height"#. Every base is #Delta x# and the heights are above, so

Then the approximation we want is:
#R_1+R_2+R_3+R_4#

#=Delta x * f(1) + Delta x * f(2) +Delta x * f(3) +Delta x * f(4)#

#=1*1+1*sqrt2+1*sqrt3+1*2=1+sqrt2+sqrt3+2=3+sqrt2+sqrt3#

#3+sqrt2+sqrt3~~3+1.414+1.732=6.146#