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Find the area under one arc of the cycloid \(x=r(\theta-\sin \theta), y=r(1-\cos \theta)\)

Area \(=\int y d x \quad y=r(1-\cos \theta) \qquad 0 \leq \theta \leq 2 \pi\)

\(d x=r(1-\cos \theta)\)

Area \(=\int_{0}^{2 \pi} y d x=\int_{0}^{2 \pi} r(1-\cos \theta) \cdot r(1-\cos \theta)d \theta\)

\(=\int_{0}^{2 \pi} r^{2}(1-\cos \theta)^{2} d \theta=r^{2} \int_{0}^{2 \pi}(1-\cos \theta)^{2} d \theta\)

Area \(=r^{2} \int_{0}^{2 \pi}(1-2 \cos \theta+\cos^{2} \theta) d \theta\)

\(=r^{2} \int_{0}^{2\pi} 1-2 \cos \theta+ \left(\frac{1}{2}+\frac{1}{2} \cos (2 \theta)\right) d \theta\)

Area \(=r^{2} \int_{0}^{2 \pi}\left(\frac{3}{2}-2 \cos \theta+\frac{1}{2} \cos (2 \theta)\right) d \theta\)

\(=r^{2}\left[\frac{3}{2} \theta-2 \sin \theta+\frac{1}{2} \cdot \frac{1}{2} \cdot \sin (2 \theta)\right]_{0}^{2 \pi}\)

\(=r^{2}\left[\frac{3}{2} \theta-2 \sin \theta+\frac{1}{4} \sin 2 \theta\right]_{0}^{2 \pi}\)

\(=r^{2}\left[\frac{3}{2}(2 \pi)-0+0-0\right]=r^{2} \cdot \frac{3}{2}(2 \pi)=3 \pi r^{2}\)

Find the arc length of the curve defined by the parametric equations:

\(x=e^{t} \cos (t),\quad y=e^{t} \sin (t), \quad 0 \leq t \leq \pi\)

Arc length \(=\int_{0}^{\pi} \sqrt{\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}} d t\)

\(\frac{d x}{d t}=\frac{d\left(e^{t} \cos t\right)}{d t}=e^{t} \cdot \cos t+-\sin (t) \cdot e^{t}=e^{t} \cos t-e^{t} \sin t=e^{t}(\cos t-\sin t)\)

\(\frac{d y}{d t}=\frac{d\left(e^{t} \sin t\right)}{d t}=e^{t} \cdot \sin t+\cos (t) \cdot e^{t}=e^{t} \cos t+e^{t} \sin (t)=e^{t}(\cos t+\sin (t))\)

\(\left(\frac{d x}{d t}\right)^{2}=\left(e^{t}(\cos t-\sin t)\right)^{2}=e^{2 t}(\cos t- \sin t)^{2}=e^{2 t}\left(\cos ^{2} t-2 \cos t \sin t+\sin ^{2} t\right)\)

\(\left(\frac{d y}{d t}\right)^{2}=\left(e^{t}(\cos t+\sin t)\right)^{2}=e^{2 t}(\cos t+\sin t)^{2}=e^{2 t}\left(\cos ^{2} t+2 \cos \sin t+\sin ^{2} t\right)\)

\(\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}=e^{2 t}\left(\cos ^{2} t-2 \cos t \sin t+\sin ^{2} t\right)+e^{2 t}\left(\cos ^{2} t+2 \cos t \sin t+\sin ^{2} t\right)\)

\(=e^{2t}\left(\cos ^{2} t-2 \cos ^{2} \sin t+\sin ^{2} t+\cos ^{2} t+2 \cos t \sin t+\sin ^{2} t\right)=e^{2t} (2)\)

Arc length \(=\int_{0}^{\pi} \sqrt{\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}} d t\)

\(=\int_{0}^{\pi} \sqrt{2 e^{2 t}} d t=\int_{0}^{\pi} \sqrt{2} \cdot \sqrt{e^{2t}} d t\)

\(=\sqrt{2} \int_{0}^{\pi} e^{t} d t=\sqrt{2}\left[e^{t}\right]_{0}^{\pi}\)

\(=\sqrt{2}\left[\begin{array}{ll}{e^{\pi}} & {-e^{0}}\end{array}\right]\)

\(=\sqrt{2}\left[e^{\pi}-1\right]\)

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