Question

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?

Answer 1

`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`