How do you use the trapezoidal rule with n=6 to approximate the area between the curve #6sqrt(lnx)# from 1 to 4?

1 Answer
Oct 25, 2015

Answer:

See the explanation section, below.

Explanation:

To approximate the Integral #int_a^b f(x) dx# using trapezoidal approximation with #n# intervals, use
#T_n=(Deltax)/2 [f(x_0)+2f(x_1)+2f(x_2)+ * * * 2f(x_(n-1))+f(x_n)] #

In this question we have:
#f(x) = 6sqrt(lnx)#
#{a,b] = [1, 4]#, and
#n=6#.

So we get
#Delta x = (b-a)/n = (4-1)/6 = 1/2 = 0.5#

The endpoints of the subintervals are found by beginning at #a=1# and successively adding #Delta x = 0.5# to find the points until we get to #x_n = b = 4#.

#x_0 = 1#, #x_1 = 1.5#, #x_2 = 2#, #x_3 = 2.5#, #x_4 = 3#, #x_5 = 3.5#, #x_6 = 4=b#,

Now apply the formula (do the arithmetic) for #f(x) = 6sqrt(lnx)#.

#T_6=(Deltax)/2 [f(x_0)+2f(x_1)+2f(x_2)+ * * * 2f(x_5)+f(x_6)] #