calculating area under curve

Question

if you calculate the area under the curve f(x)=x² from x=0 to x=5 using a left had rectangular approximation, a right hand approximation, and a midpoint rectangular approximation, in each case using interval widths of 1, which of the following will be true?

 

a) the result of the left hand approximation will be lager than the result for the right hand approximation

b) the result of the midpoint approximation will be lager than the result for the left hand approximation

c) of the three, the result of the midpoint approximation will be the largest

Answer 1

 

 

The total interval is divided into 5 subintervals of width 1 each

 

The five ordinates are corresponding to x=0,1,2,3,4

Corresponding y ordinate =0,1,4,9,16

 

Integral = o*1+1*1+4*1+9*1+16*1

             =30

 

Using right hand approximation--------

 

x=1,2,3,4,5

y=1,4,9,16,25

 

Integral =1*1+4*1+9*1+16*1+25*1

             =55

 

Using mid point approximation -----------

 

x=0.5,1.5,2.5,3.5,4.5

y=0.25,2.25,6.25,12.25,20.25

 

Integral =0.25*1+2.25*1+6.25*1+12.25*1+20.25*1

            =41.25

 

Answer b is correct