ESTIMATE THE AREA UNDER THE GRAPH

Question

Estimate the area under the graph of f(x)=1/(x+4) over the interval [1,3] using 5 approximating rectangles and right end points.

Answer 1

y = 1/(x+4) = (x+4)^-1

integral of y = ln(x+4)

evaluating from 1 to 3 =

ln(3+4) - ln(1+4) = ln7-ln5 = ln(7/5) = ln1.4 = about .3365 = area under the curve from 1 to 3

any approximation by rectangles should be near that number

5 rectangles from 1 to 3 means each has a base of (3-1)/5 = 2/5

right end points would be 7/5, 9/5, 11/5, 13/5, and 15/5 = 5

corresponding heights = f(7/5) = 1/(7/5+4) =1/(7/5+20/5) = 1/27/5 = 5/27

f(9/5) = 1/(9/5+4) = 1/(9/5+20/5) = 1/29/5 = 5/29

f(11/5) = 5/31

f(13/5) = 5/33

f(15/5) = 5/35 = 1/7

sum of heights = 5/27+5/29+5/31+5/33+ 5/35 = about .1852 + .1724 + .1613 + .1515 + .1429

=0.8133

multiply by 2/5 or 0.4

= .8133x .4 = 0.3253 which is close to .3365