6.3 Summation Notation

Question

The expression (1)/(5)(ln(2+(1)/(5))+ln(2+(2)/(5))+ln(2+(3)/(5))+ln(2+(4)/(5))+ln(2+(5)/(5))) is a Riemann sum approximation of which of the following integrals?

a) \int_2^3 ln((x)/(5))dx

b) (1)/(5)\int_0^5 ln x dx

c) (1)/(5)\int_2^3 ln x dx

d) \int_2^3 ln x dx

e) \int_0^5 ln(2+x)dx

Answer 1

d) \int_2^3 ln x dx

you can imagine the approximation as counting the areas of small rectangles under the given function. Each box has a with of 1/5 and a height ranging from ln(2+1/5) to ln(3). This is the right handed Riemann sum approximation for the function ln(x) between x=2 and x=3.

by looking at the evaluation of the sum... the equation can be rewritten as:

(1/5)*( ln(2.2) + ln(2.4) + ln(2.6) + ln(2.8) + ln(3.0) )

or

(1/5)*ln(2.2) + (1/5)*ln(2.4) + (1/5)*ln(2.6) + (1/5)*ln(2.8) + (1/5)*ln(3.0) )

from here you can see the similarities between this summation and the function ln(x). Specifically this equation computes approximate the right hand riemann sum with 5 boxes, each with a width of 1/5 and a height ranging from ln(2) to the ln(3)... this leads us to our final answer that this summ approximates the integral \int_2^3 ln x dx