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It is often harder to find areas bounded by curves than straight lines.

For example...

Regardless of guessing skills, this is a tough question to answer without calculus. (Calculus makes it easy- Integrate $y=x^2$ to get $\frac{x^3}{3}$, plug in $x=1$, and you win!)

Method 1: There is a well-known formula that the area of a parabolic sector (parabola cut off by a line) is $\frac{2}{3} bh$. B is the base, or the length of the line segment inside the parabola, and h is the height.

If we take the parabola $y=x^2$, and let the line by $y=x$, then obviously the base is $\sqrt{2}$. We can find the height is $\frac{\sqrt{2}}{4}$, so the area is $\frac{1}{3}$.