The curve above is drawn in the $x y$-plane and is described by the equation in polar coordinates $r= heta+\sin (2 heta)$ for $0 \leq heta \leq \pi$, where $r$ is measured in meters and $ heta$ is measured in radians. The derivative of $r$ with respect to $ heta$ is given by $\frac{d r}{d heta}=1+2 \cos (2 heta)$.
(a) Find the area bounded by the curve and the $x$-axis.
(b) Find the angle $ heta$ that corresponds to the point on the curve with $x$-coordinate -2 .
(c) For $\frac{\pi}{3}

Question

The curve above is drawn in the $x y$-plane and is described by the equation in polar coordinates $r=	heta+\sin (2 	heta)$ for $0 \leq 	heta \leq \pi$, where $r$ is measured in meters and $	heta$ is measured in radians. The derivative of $r$ with respect to $	heta$ is given by $\frac{d r}{d 	heta}=1+2 \cos (2 	heta)$.
(a) Find the area bounded by the curve and the $x$-axis.
(b) Find the angle $	heta$ that corresponds to the point on the curve with $x$-coordinate -2 .
(c) For $\frac{\pi}{3}<	heta<\frac{2 \pi}{3}, \frac{d r}{d 	heta}$ is negative. What does this fact say about $r$ ? What does this fact say about the curve?
(d) Find the value of $	heta$ in the interval $0 \leq 	heta \leq \frac{\pi}{2}$ that corresponds to the point on the curve in the first quadrant with greatest distance from the origin. Justify your answer.

Accepted Answer

∻Solution by Steps∻ ### (a) Find the area bounded by the curve and the $x$-axis. ‖ step 1 ⋮ The area bounded by the curve and the $x$-axis in polar coordinates is given by the integral: $\frac{1}{2} \int_{0}^{\pi} r^2 \, d heta$ ‖ ‖ step 2 ⋮ Substitute $r = heta + \sin(2 heta)$ into the integral: $\frac{1}{2} \int_{0}^{\pi} ( heta + \sin(2 heta))^2 \, d heta$ ‖ ‖ step 3 ⋮ Expand the integrand: $\frac{1}{2} \int_{0}^{\pi} ( heta^2 + 2 heta \sin(2 heta) + \sin^2(2 heta)) \, d heta$ ‖ ‖ step 4 ⋮ Integrate each term separately: $\frac{1}{2} \left[ \int_{0}^{\pi} heta^2 \, d heta + 2 \int_{0}^{\pi} heta \sin(2 heta) \, d heta + \int_{0}^{\pi} \sin^2(2 heta) \, d heta ight]$ ‖ ‖ step 5 ⋮ Evaluate the integrals: $\frac{1}{2} \left[ \frac{\pi^3}{3} + 0 + \frac{\pi}{2} ight] = \frac{1}{2} \left( \frac{\pi^3}{3} + \frac{\pi}{2} ight) = \frac{\pi^3}{6} + \frac{\pi}{4}$ ‖ ### (b) Find the angle $ heta$ that corresponds to the point on the curve with $x$-coordinate -2. ‖ step 1 ⋮ The $x$-coordinate in polar coordinates is given by $x = r \cos( heta)$ ‖ ‖ step 2 ⋮ Substitute $r = heta + \sin(2 heta)$ and $x = -2$: $-2 = ( heta + \sin(2 heta)) \cos( heta)$ ‖ ‖ step 3 ⋮ Solve the equation $-2 = ( heta + \sin(2 heta)) \cos( heta)$ for $ heta$ ‖ ### (c) For $\frac{\pi}{3} < heta < \frac{2\pi}{3}$, $\frac{d r}{d heta}$ is negative. What does this fact say about $r$? What does this fact say about the curve? ‖ step 1 ⋮ Given $\frac{d r}{d heta} = 1 + 2 \cos(2 heta)$ ‖ ‖ step 2 ⋮ For $\frac{\pi}{3} < heta < \frac{2\pi}{3}$, $\cos(2 heta)$ is negative, making $1 + 2 \cos(2 heta)$ negative ‖ ‖ step 3 ⋮ Since $\frac{d r}{d heta}$ is negative, $r$ is decreasing in this interval ‖ ‖ step 4 ⋮ The curve is moving closer to the origin as $ heta$ increases from $\frac{\pi}{3}$ to $\frac{2\pi}{3}$ ‖ ### (d) Find the value of $ heta$ in the interval $0 \leq heta \leq \frac{\pi}{2}$ that corresponds to the point on the curve in the first quadrant with greatest distance from the origin. Justify your answer. ‖ step 1 ⋮ The distance from the origin in polar coordinates is given by $r = heta + \sin(2 heta)$ ‖ ‖ step 2 ⋮ To find the maximum distance, take the derivative and set it to zero: $\frac{d r}{d heta} = 1 + 2 \cos(2 heta) = 0$ ‖ ‖ step 3 ⋮ Solve for $ heta$: $1 + 2 \cos(2 heta) = 0 \Rightarrow \cos(2 heta) = -\frac{1}{2} \Rightarrow 2 heta = \frac{2\pi}{3} \Rightarrow heta = \frac{\pi}{3}$ ‖ ‖ step 4 ⋮ Verify that $ heta = \frac{\pi}{3}$ is within the interval $0 \leq heta \leq \frac{\pi}{2}$ and corresponds to a maximum by checking the second derivative or evaluating $r$ at endpoints ‖ ∻Answer∻ ⚹ (a) $\frac{\pi^3}{6} + \frac{\pi}{4}$ ⚹ ⚹ (b) Solve $-2 = ( heta + \sin(2 heta)) \cos( heta)$ for $ heta$ ⚹ ⚹ (c) $r$ is decreasing; the curve moves closer to the origin ⚹ ⚹ (d) $ heta = \frac{\pi}{3}$ ⚹ ∻Key Concept∻ ⚹Polar coordinates and integrals⚹ ∻Explanation∻ ⚹The problem involves finding areas, solving equations, and analyzing behavior in polar coordinates using calculus techniques.⚹◊What is the significance of the derivative $\frac{d r}{d heta}=1+2 \cos (2 heta)$ in analyzing the given polar curve $r= heta+\sin (2 heta)$⍭ Generate me a similar question◊

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