This question refers to the function $f(x)=\sqrt[\phantom{2}]{144-z^2}$

a$.$ Graph this function using Desmos. What is the domain of this function?

b$.$ Estimate the definite integral ${\displaystyle \underset{-12}{\overset{12}{}}}f(x)dx$ usirg a left endpoint Piemann sum with $n=8$ boxes.

How did you determine $x$ for this calculation?

Draw a graph of the function $($this can be a shetch or a digital picturel that shows the eight left endocint bexes used in your sum.

Show the table of function values that you used, and shore your work for the calculation.

Do you think that this approximation is an over$-$ or under$-$estimate of the true value of ${\displaystyle \underset{-12}{\overset{12}{}}}f(x)dx?$

c$.$ Now estimate the definite integral ${\displaystyle \underset{-12}{\overset{12}{}}}f(x)dx$ using a midpoint Riemann sum with $n=8$ boxes

How did you determine $x$ for this calculation?

Draw a graph of the function $($this can be a sketch or a digital picturel that shows the eight midpoint bowes used in your sum.

Show the table of function values that you used, and show your work for the calculation. What is alferent about the table tha you needed for this calculation as apposed to the previous calculation?

Do you think that this approximation is an over$-$ or under$-$estimate of the true value of ${\displaystyle \underset{-12}{\overset{12}{}}}f(x)dx?$

d$.$ The graph of this function is a familier shape. Use your knowledge of geometry to fgure out the exact value of ${\displaystyle \underset{-12}{\overset{12}{}}}f(x)d$x$.$ What was the error in your two approximations above?