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

1 Answer
Mar 24, 2015

For this problem: f(x)=sqrtxf(x)=x

a=0a=0 and b=4b=4

the number of rectangles =n=4=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