Question

How do you find the error that occurs when the area between the curve `y=x^3+1` and the x-axis over the interval [0,1] is approximated by the trapezoid rule with n = 4?

Answer 1

Assuming that you want the actual error, use the fundamental theorem of calculus to get the exact value of `int_0^1(x^3+1)dx` Then find the difference (subtract) from the trapezoid approximation.

I guess it's easy to see that
`int_0^1(x^3+1)dx=(x^4/4+x)]_0^1=(1/4+1)-(0)=4/5=1.25`.
The trapezoid rule with `n=4` gives `1.265625`

The error, then is `0.015625`.