$4.$ A geometric integral $($a$)$ Consider the function g$($x$)=$ Z x $0$ p $1$ t $2$ dt$.$ Suppose $0$ x $1.$ Use geometry to find a formula for g$($x$).($Hint: partition the area under the curve into a sector of a circle and a triangle, as in the image below $($Hint $2$: your answer may involve an inverse trig function$)).$ Show your work carefully. $($b$)$ Briefly explain why the formula you found in a$)$ also works if $1$ x $0,$ so in fact it is true for all x in the domain of g$.($c$)$ What does the fundamental theorem tell you $($without doing any calculation$)$ about g $($x$)?($d$)$ Now calculate g $($x$)$ by directly differentiating the answer you found in $($a$)$ and check that it matches your answer from $($c$).$ Congratulations! You used geometry to find an interesting antiderivative.