SOLUTION: A rectangle is bounded by the x-axis and the semicircle y=[36-x^2]square root. Write the area of the rectangle as a function of x, and determine the domain of the function

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Question 162453: A rectangle is bounded by the x-axis and the semicircle y=[36-x^2]square root. Write the area of the rectangle as a function of x, and determine the domain of the function.
This is the picture of the graph: http://hmco.tdlc.com/pre3e/common/ch30/ch30c/30c_images/2901087.gif

Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
A rectangle is bounded by the x-axis and the semicircle . Write the area of the rectangle as a function of x, and determine the domain of the function.
This is the picture of the graph: http://hmco.tdlc.com/pre3e/common/ch30/ch30c/30c_images/2901087.gif

Label the upper right hand corner of the
rectangle (x,y):



Now if we just look at the right half of that
rectangle, we have:



The area of the right half of the rectangle
is .

Therefore the area of the entire rectangle:



is twice that or 

.

However, that is the area in terms of both
x and y.  

We must get the area A in terms of just x.

So we replace  in  by 



Your teacher may want you to write  instead of 

If so the answer in functional notation is: 



Edwin

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