Question

Using the Riemann Summ Estimate ln(4)

a) n = 3

b) n = 4

c) n = 5

Answer 1

f(x)=1/x

the domain of x is[1,4]

when n=3

=(4-1)/3=1

subintervals are [1,2],[2,3],[3,4]

left end points are 1,2,3

left rectangle approximation=(1+1/2+1/3)x1=11/6

right end points are 2,3,4

right rectangle area =(1/2+1/3+1/4)x1=26/24=13/12

mid ponts are3/2,5/2,7/2

mid rectangle area=(2/3+2/5+2/7)=142/105

when n=4

=(4-1)/4=3/4

subintervals are [1,7/4],[7/4,5/2],[5/2,13/4],[13/4,4]

left end points are 1,7/4,5/2,13/4

left rectangle approximation=(1+4/7+2/5+4/13)x3/4=1.7

right end points are 7/4,5/2,13/4,4

right rectangle area =(4/7+2/5+4/13+1/4)x3/4=3(1/7+1/10+1/13+1/16)

mid ponts are11/8,17/8,23/8,39/8

mid rectangle area=(8/11+8/17+8/23+8/39)x3/4=2(1/11+1/17+1/23+1/39)

when n=5

=(4-1)/5=3/5

subintervals are [1,8/5],[8/5,11/5],[11/5,14/5],[14/5,17/5],[17/5,4]

left end points are 1,8/5,11/5,14/5,17/5

left rectangle approximation=(1+5/8+5/11+5/14+5/17)x3/5=3(1/5+1/8+1/11+1/14+1/17)

right end points are 8/5,11/5,14/5,17/5,4

right rectangle area =(5/8+5/11+5/14+5/17+1/4)x3/5=3(1/8+1/11+1/14+1/17+1/20)

mid ponts are13/10,19/10,25/10,31/10,37/10

mid rectangle area=(10/13+10/19+10/25+10/31+10/37)x3/5=6(1/13+1/19+1/25+1/31+1/37)