What is the convergence order of the numerical formula?

Question

$\int_0^2 e^x \cos (x) \sin (x) dx$,

What is the convergence order of the numerical formula?

I am confusing the question.

Do I have to use Trapezoidal rule or simpson rule or approximation rule?

If so then how to use approximation and finding order of convergence?

Help me

Answer 1

If you want to use a numerical method like the trapezoidal rule, based on a decomposition $0=x_0, x_1, \cdots, x_n=2$ of equally space points (spacing $h$), you just need to compute

$$ \frac h2 \left(f(x_0) + f(x_n) + 2 \sum_{i=1}^{n-1} f(x_i) \right), $$

where $f(x)=\sin x \cos x e^x$. Regarding the order of convergence, you have that the absolute error is given by

$$ \left|\frac{(b-a) h^2}{12} f''(\xi)\right|, \quad \xi \in [0,2] $$

so, if the second derivative of $f$ is bounded (which is the case) the error will behave like $h^2$.

Answer 2

You can do this integral by doing integration by parts twice with $\cos x \sin x = \frac{1}{2}\sin 2x$. This will help you check your answer.