Question

How do you use the trapezoidal rule with n=4 to approximate the area between the curve `y(t)=(t^3 +t)` from 0 to 2?

Answer 1

Refer to explanation

Explanation:

Hence we have n=4 we got 4 trapezoid hence we have that

`Δx=(b-a)/n=(2-0)/4=1/2`

From the trapezoidal rule we have that

`int_0^2 (t^3+t)dt=(f(0)+f(1/2))/2*Δx+(f(1/2)+f(1))/2*Δx+(f(1)+f(3/2))/2*Δx+(f(3/2)+f(2))/2*Δx=(Δx)/2*[f(0)+2f(1/2)+2f(1)+2f(3/2)+f(2)]`

Hence `f(0)=0`
`f(1/2)=5/8`
`f(1)=2`
`f(3/2)=39/8`
`f(2)=10`

So we have that

`int_0^2 (t^3+t)dt=(Δx)/2*[f(0)+2f(1/2)+2f(1)+2f(3/2)+f(2)]=(1/2)/2*[0+2*5/8+2*2+2*39/8+10]=1/4*(25)=6.25`