Area under the curve

Question

suppose we have graph of $\sin(x)$ and a ellipse say $\frac{x^2}{10}+\frac{y^2}{2}=1$ it comes like following

now when we intregrate $\int_{0}^{2\pi}\sin(x)dx$ it comes out to be zero , because area above $x$-axis is equal to area below $x$-axis so therefore when we compute area of ellipse it should also be equal to zero ? why is't it taken to be zero?

Answer 1

As The Chaz said, computing the integral gives a signed integral; so, since your ellipse is centered, the result is zero. If you move the center of the same ellipse up or down, you will get a different result.