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?

Question

Charles Anctil · Accepted Answer

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#

Charles Anctil · Answer

undefined To estimate the area under the graph of $ f(x) = \sqrt{x} $ from $ x = 0 $ to $ x = 4 $ using four approximating rectangles and right endpoints, follow these steps:

1. Divide the interval $[0, 4]$ into four equal subintervals. Since there are four rectangles, each subinterval will have a width of $ \Delta x = \frac{4 - 0}{4} = 1 $.

2. Choose right endpoints within each subinterval. For the first subinterval, $ x_1 = 1 $, for the second, $ x_2 = 2 $, for the third, $ x_3 = 3 $, and for the fourth, $ x_4 = 4 $.

3. Calculate the function values at each right endpoint. Evaluate $ f(x) = \sqrt{x} $ at $ x_1 = 1 $, $ x_2 = 2 $, $ x_3 = 3 $, and $ x_4 = 4 $. This gives $ f(1) = 1 $, $ f(2) = \sqrt{2} $, $ f(3) = \sqrt{3} $, and $ f(4) = 2 $.

4. Now, each rectangle's height corresponds to the function value at the right endpoint of its respective subinterval.

5. Calculate the area of each rectangle by multiplying its height by its width. The area of the first rectangle is $ 1 \times 1 $, the second is $ \sqrt{2} \times 1 $, the third is $ \sqrt{3} \times 1 $, and the fourth is $ 2 \times 1 $.

6. Add up the areas of all four rectangles to get an approximation of the total area under the curve.

$$ \text{Area} \approx 1 + \sqrt{2} + \sqrt{3} + 2 $$