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How do you use the trapezoidal rule with n=4 to approximate the area between the curve #y=sqrt(x+1)# from 1 to 3?

Answer 1

#"Area"~=3.4439" square units"#

The trapezoidal rule says that the area can be found using the formula: #"A"=h/2(y_1+y_n+2(y_2+y_3+...+y_(n-1)))#

#A# is the area, #h# is the step length(the length of a single strip), #y_1,y_2,...,y_n# are the #y# values corresponding to each #x# value taken from the interval #[1,3]#

In this problem, since #n=4#, #A=h/2(y_1+y_4+2(y_2+y_3))#

First, #h=(x_n-x_1)/(n-1)#

#x_1# and #x_n# are respectively the first and last #x# values in the interval

#h=(3-1)/(4-1)=2/3#

#y_1=sqrt(x_1+1)=sqrt(1+1)=sqrt2# #y_2=sqrt(x_2+1)=sqrt((1+2/3)+1)=sqrt(5/3+1)=sqrt(8/3)# #y_3=sqrt(x_3+1)=sqrt((5/3+2/3)+1)=sqrt(7/3+1)=sqrt(10/3)# #y_4=sqrt(x_4+1)=sqrt((7/3+2/3)+1)=sqrt(3+1)=sqrt4=2#

NB : In order to get the next #x# we add #2/3# to the previous. Example : #x_2=x_1+2/3=1+2/3=5/3#

Finally,

#A=(2/3)/2(sqrt2+2+2(sqrt(8/3)+sqrt(10/3)))~=color(blue)(3.4439)#

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.