When finding the area under a curve, there appears to be a contradiction
Like area under integral sinx from 0 to 2pi is zero because, the areas above and below cancel each other.
But when finding the area under a question like one given below that doesn't seem to be the case, Why is that so?
Shouldn't this area be zero since its also symmetrically occupied above and below x axis(just like the case of sinx)?
On
Just to add, when we integrate we integrate a function y = f(x), here x can be anything, y can be anything and there is a good reason we take the sign of y as it is. Suppose x indicates time and y indicates flow rate of water in a reservoir +y indicates inflow rate and -y indicates outflow rate, and some how y obeys $y =C\sin{x}$ where C is a constant and x is time, if you want to to know (and that will be the interest of almost all) how much water is left in the reservoir after $2{\pi}$ unit of time you have to take the sign as it is as the result is zero.
In the real world sign of y has implications, it is a variable and it can indicate anything, so is x and so is integration of y.
But when you are asked to find "area bounded by curves" we can safely assume the questioner is interested only in areas, does not bother about sign of y.
When asked to find an area, you always have to know whether the current interpretation of "area" allows for areas to "cancel out", as you say.
Some times the context allows for this cancellation, such as when finding $$ \int_0^{2\pi}\sin x\,dx = 0 $$ and interpreting that as a statement about areas.
However, some times (such as when asked to find the area of a region of the plane), areas aren't allowed to cancel out. They only ever add up.
Basically, integrals can cancel, but areas never can. You will have to use your best judgement on a case-by-case basis to figure out which interpretation is relevant.