How do you use the trapezoidal rule with n=9 to approximate the area between the curve #y=x^2 -2x +2# from 0 to 3?

1 Answer
Sep 14, 2015

Answer:

We use the formula #1/6[f(x_0)+2f(x_1)+2f(x_2)+ cdots+ 2f(x_8) + f(x_9)]# where #f(x) = x^2 -2x +2# and #x_0 cdots x_9# are chosen to evenly divide the interval #[0,3]# into 9 pieces.

Explanation:

The trapezoidal rule works by dividing the area under the curve into #n# trapezoids, then calculating the area of each trapezoid, and summing them up. The formula is:

#int_a^b f(x)dx = (b-a)/(2n)[f(x_0)+2f(x_1)+2f(x_2)+cdots2f(x_{n-1})+f(x_n)]#

In our case, #n=9#, #a = 0#, and #b=3#, so we have:

#int_0^3 f(x)dx = (3-0)/(2*9)[f(x_0)+2f(x_1)+2f(x_2)+2f(x_3)+2f(x_4)+2f(x_5)+2f(x_6) + 2f(x_7) + 2f(x_8) + f(x_9)]#

We choose the #x_0 cdots x_9# so that they divide up the interval from 0 to 3 evenly. So #x_0 = 0, x_1 = 1/3, x_2 = 2/3, x_3 = 1, x_4 = 4/3, x_5 = 5/3, x_6 = 2, x_7 = 7/3, x_8 = 8/3, x_9 = 9#.

From there it's just a matter of plugging these numbers into #f(x)# and solving the formula.