# Area under the curve

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?

• The integral gives signed area. Also, what's the function for the ellipse? Nov 1 '13 at 5:54
• for half-upper part $f(x)=\sqrt{2-\frac{2x^2}{10}}$ Nov 1 '13 at 5:57
• The ellipse above can be described by two functions, say $f_1$ and $f_2 = - f_1$. The area is given by $\int (f_1-f_2) = 2 \int f_1$. For the $\sin$ function, we have $f_1 = \sin$, $f_2 = 0$. Nov 1 '13 at 5:59