009C Sample Final 1, Problem 8 Detailed Solution

A curve is given in polar coordinates by

$r=1+\sin 2\theta$

$0\leq \theta \leq 2\pi$

(a) Sketch the curve.

(b) Find the area enclosed by the curve.

---

Background Information:
The area under a polar curve  Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle r=f(\theta )}   is given by
$\int _{\alpha _{1}}^{\alpha _{2}}{\frac {1}{2}}r^{2}~d\theta$  for appropriate values of  $\alpha _{1},\alpha _{2}.$

Solution:

(a)
450px

(b)

Step 1:
Since the graph has symmetry (as seen in the previous image),
the area of the curve is
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A=2\int _{-{\frac {\pi }{4}}}^{\frac {3\pi }{4}}{\frac {1}{2}}(1+\sin(2\theta ))^{2}~d\theta .}

Step 2:
Using the double angle formula for  $\sin(2\theta ),$  we have
${\begin{array}{rcl}\displaystyle {A}&=&\displaystyle {\int _{-{\frac {\pi }{4}}}^{\frac {3\pi }{4}}1+2\sin(2\theta )+\sin ^{2}(2\theta )~d\theta }\\&&\\&=&\displaystyle {\int _{-{\frac {\pi }{4}}}^{\frac {3\pi }{4}}1+2\sin(2\theta )+{\frac {1-\cos(4\theta )}{2}}~d\theta }\\&&\\&=&\displaystyle {\int _{-{\frac {\pi }{4}}}^{\frac {3\pi }{4}}{\frac {3}{2}}+2\sin(2\theta )-{\frac {\cos(4\theta )}{2}}~d\theta }\\&&\\&=&\displaystyle {{\frac {3}{2}}\theta -\cos(2\theta )-{\frac {\sin(4\theta )}{8}}{\bigg |}_{-{\frac {\pi }{4}}}^{\frac {3\pi }{4}}.}\\\end{array}}$

Step 3:
Lastly, we evaluate to get
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {A}&=&\displaystyle {{\frac {3}{2}}{\bigg (}{\frac {3\pi }{4}}{\bigg )}-\cos {\bigg (}{\frac {3\pi }{2}}{\bigg )}-{\frac {\sin(3\pi )}{8}}-{\bigg [}{\frac {3}{2}}{\bigg (}-{\frac {\pi }{4}}{\bigg )}-\cos {\bigg (}-{\frac {\pi }{2}}{\bigg )}-{\frac {\sin(-\pi )}{8}}{\bigg ]}}\\&&\\&=&\displaystyle {{\frac {9\pi }{8}}+{\frac {3\pi }{8}}}\\&&\\&=&\displaystyle {{\frac {3\pi }{2}}.}\\\end{array}}}

Final Answer:
(a)     See above.
(b)     Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\frac {3\pi }{2}}}

Return to Sample Exam