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Activity 12.5.3 .
Consider the circle given by
\(x^2 + (y-1)^2 = 1\) as shown in
Figure 12.5.12 .
Figure 12.5.11.
(a) Determine a polar curve in the form
\(r = f(\theta)\) that traces out the circle
\(x^2 + (y-1)^2 = 1\text{.}\) You should substitute
\(x = r \cos(\theta)\) and
\(y = r \sin(\theta)\) into the rectangular coordinate equation and solve for
\(r\text{.}\)
Hint .
After you substitute, you should expand and combine like terms. You will also want to factor your expression and disregard the
\(r=0\) solution.
(b) Find the exact average value of
\(g(x,y) = \sqrt{x^2 + y^2}\) over the interior of the circle
\(x^2 + (y-1)^2 = 1\text{.}\)
Hint .
Use a radially simple description of your region with your result from part a as the upper bound.
(c) Find the volume under the surface
\(h(x,y) = x\) over the region
\(D\text{,}\) where
\(D\) is the region bounded above by the line
\(y=x\) and below by the circle (this is the shaded region in
Figure 12.5.12 ).
Figure 12.5.12.
(d) Explain why in both parts (b) and (c) it is advantageous to use polar coordinates.
