The line of cycloids is set parametrically. When it comes to the first arch, then there is a limitation:
0<0<2π and then the first arch is drawn when the parameter value changes within: 0≤t≤2π. We use the formula for finding the body volume formed by a line given parametrically: V=π∫abf2(t)x′(t)dt. Substitute ve into the formula and get: V=π∫02π(a(1−cosθ)(a(θ+sinθ)′dθ=π∫02π(a(1−cosθ)(θ+cosθ)dθ=π∫02π(aθ−acos2θ+acosθ−axcosθ)dθ=−aπ∫02π(21cos2θ+21)dθ−aπ∫02πθcosθdθ+aπ∫02πcosθdθ+aπ∫02πθdθ=π2(2π−1)a
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