Basic Leaf

The function $f(x)$ is defined by the equation $f(x)=\sqrt { 1-{ x }^{ 2 } }$ , or a semicircle centered at the origin with positive $f(x)$ and a radius of 1.

Define ${ g }_{ 0 }(x)=\int _{ 0 }^{ x }{ f(u)\, du }$ , ${ g }(x)={ g }_{ 0 }(x)-{ g }_{ 0 }(-1)$ , $h_{ 0 }(x)=\int _{ 0 }^{ x }{ g(u)\, du }$ , and ${ h }(x)={ h }_{ 0 }(x)-{ h }_{ 0 }(-1)$ .

Together, the graphs of $g(x)$ an $h(x)$ create a "leaf" in the coordinate plane. Find the area of this leaf, rounded to 3 decimal places. If this leaf doesn't exist (open leaf), use -0.5 as your answer.

Hint: Try graphing the integrals.