How do you use the trapezoidal rule with n=4 to approximate the area between the curve #x ln(x+1)# from 0 to 2?

1 Answer
Aug 13, 2018

Answer:

#1/2ln6~~0.9#

Explanation:

The trapezoidal rule states that the area under an integral can be approximated by the equation:

#int_a^bf(x) \ dx~~(Deltax)/2[f(x_0)+2f(x_1)+2f(x_2)+2f(x_3)+...+2f(x_(n-1))+f(x_n)]#

where:

  • #Deltax=(b-a)/n#

  • #n# is the number of trapezoids

  • #x_0=a#

  • #x_1,x_2,...,x_n# are equally spaced #x#-coordinates of the right edges of trapezoids #1,2,3,...,n#.

So, we get:

#int_0^2xln(x+1) \ dx~~(b-a)/(2n)[f(0)+2f(1)+f(2)]#

#=(2-0)/(2*4)[f(0)+2f(1)+f(2)]#

#=2/8[0ln1+2(1ln2)+2ln3]#

#=1/4(2ln2+2ln3)#

#=1/2ln6#

#~~0.9#