Polar coordinates

Calculus Level 2

If the area of the region enclosed by $r = 7\cos 3\theta$ in the interval $(0\leq \theta \leq 2\pi) is ( \dfrac{a^2 \pi}b$ ), where $a$ and $b$ are positive integers with $b$ minimized, what is the value of $a+b$ .

10 4 5 11

1 solution

Hana Wehbi
May 21, 2016

$A=\frac{1}{2}$ $\int_{0}^{2\pi}$ $(7cos3\theta)^{2} d\theta$ =

$\frac{49}{2}$ $\int_{0}^{2\pi}$ $\frac{1+cos6(\theta)}{2}$ $d\theta$ =

= $\frac{49}{4}$ ( $\theta$ + $\frac{\sin6\theta}{6}$ ); $\theta$ from $0\to 2\pi$

= $\frac{49\pi}{4}$ ; Thus $a^{2}=49, \text {and} b=4\implies a+b=11$

Polar coordinates

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