Question

How do you Use the trapezoidal rule with `n=6` to approximate the integral `int_0^3dx/(1+x^2+x^4)dx`?

Answer 1

The answer is `4643/5187`.

The trapezoidal rule is just a formula. From what we are given, we have:

`a=0`
`b=3`
`n=6`
`f(x)=1/(1+x^2+x^4)`
`h=(b-a)/n=1/2`

The formula is:

`T=h/2[f(x_0)+2f(x_1)+2f(x_2)+2f(x_3)+2f(x_4)+2f(x_5)+f(x_6)]`
`f(x_0)=f(0)=1/1`
`f(x_1)=f(1/2)=1/(1+1/4+1/16)=16/21`
`f(x_2)=f(1)=1/(1+1+1)=1/3`
`f(x_3)=f(3/2)=1/(1+9/4+81/16)=16/133`
`f(x_4)=f(2)=1/(1+4+16)=1/21`
`f(x_5)=f(5/2)=1/(1+25/4+625/16)=16/741`
`f(x_6)=f(3)=1/(1+9+81)=1/91`
`T=1/4[1+32/21+2/3+32/133+2/21+32/741+1/91]`
`T=4643/5187~~.89512`

Using numeric integration on a graphing calculator, we get `T~~.89537`. So our answer is good for 3 sig figs.