009C Sample Final 3, Problem 8

A curve is given in polar coordinates by  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4+3\sin \theta}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0\leq \theta \leq 2\pi}

(a) Sketch the curve.

(b) Find the area enclosed by the curve.

Foundations:
The area under a polar curve   $r=f(\theta )$  is given by
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{\alpha_1}^{\alpha_2} \frac{1}{2}r^2~d\theta}   for appropriate values of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_1,\alpha_2.}

Solution:

(a)
Insert graph

(b)

Step 1:

Step 2:

Final Answer:
(a)
(b)

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