So I was playing with my calculator and I typed this out:
$$\int_{0}^{2\pi} \sin{(x)} dx$$
And what do I get:
$$0.3441690684$$
I guessed that's a reasonable amount of error from a machine using numeric techniques to calculate these things. Then I tried:
$$\int_{0}^{2\pi} \cos{(x)} dx$$
And get:
$$6.270599475$$
Why? Which numeric integration method leads to this erroneous answer. While integrating, which functions do I trust the calculator to give me correct answers for and which functions will tend to make me more wrong than with a paper-and-pen.
And why is the error different for sine and cosine functions? Makes no sense.
Your calculator has probably been set to degrees rather than radians
Note that (using radians in the trigonometric functions)
$$\int_0^{2\pi}\sin\left( \frac{\pi x}{180}\right) \,dx= \frac{180}{\pi}\left(1-\cos\left(\frac{2\pi^2}{180}\right)\right) \approx 0.3441690684217562$$
$$\int_0^{2\pi}\cos\left( \frac{\pi x}{180}\right) \,dx= \frac{180}{\pi}\left(\sin\left(\frac{2\pi^2}{180}\right)-0\right) \approx 6.270599474641442$$