Optimization Problems; Antiderivatives; and Areas and Distances

Question

Unless specified, all approximating rectangles are assumed to have the same width.

Let 

f(x) = 1/x

.

(a)

Estimate the area under the graph of f, the x-axis, and the lines 

x = 1

 and 

x = 2

 using four approximating rectangles and right endpoints. (Round your answer to four decimal places.)

(b)

Repeat part (a) using a left endpoints. (Round your answer to four decimal places.)

Answer 1

integral of f(x) = 1/x

= lnx + C

evaluated from 1 to 2

= ln2 - ln1 = ln2

Area = about 0.69315

the approximations with rectangles should

give slightly more than and slightly less than

the area above, 0.6932

it helps to draw a graph. it should be slightly less than .5+.25 = .75, if you construct a rectangle and triangle connecting the points (0,1),(0,2), (1,1) and (2,1/2) as vertices of a rectangle with a triangle on top.

Reiman rectangle areas= base of (1/4) times sum of heights. Left side heights overestimate the total area

Right side heights underestimate the total area

left side approximation = (1/4)(1+4/5+2/3+4/7)

= 1/4 +1/5+1/6+1/7 =about .7595

right side: (1/4)(4/5+2/3+7/4+1/2)= 1/8 less than ..7595

= 1/5+1/6+1/7+1/8

=.7595-.125 = .6345

average of the approximations = (.7595+.6345)/2 = about 0.6970

which is very close to 0.6932, off by just 0.0038