【微分積分】8-5-4 図形の重心と回転体の体積｜要点まとめ

\(D\)の面積は
\(\displaystyle |D|=\frac{\pi a^2}{2}\)
図形の対称性より\(x_G=0\)
\(\displaystyle y_G=\frac{1}{|D|}\iint_Dydxdy\)
\(\displaystyle =\frac{2}{\pi a^2}\int_{-a}^a\int_0^\sqrt{a^2-x^2}ydydx\)
\(\displaystyle =\frac{2}{\pi a^2}\int_{-a}^a\frac{a^2-x^2}{2}dx\)
\(\displaystyle =\frac{4a}{3\pi}\)
よって、重心の座標は\(\displaystyle \left(0,\frac{4a}{3\pi}\right)\)

\(D\)の面積は
\(\displaystyle |D|=\frac{1}{2}a^2・2\alpha=a^2\alpha\)
図形の対称性より\(y_G=0\)
\(D\)を極座標変換すると
\(\displaystyle E=\{(r,\theta)|0\leqq r\leqq a,-\alpha\leqq \theta\leqq \alpha\}\)
\(\displaystyle x_G=\frac{1}{|D|}\iint_Dxdxdy\)
\(\displaystyle =\frac{1}{a^2\alpha}\iint_Er\cos\theta・rdrd\theta\)
\(\displaystyle =\frac{1}{a^2\alpha}\int_0^ar^2dr\int_{-\alpha}^\alpha\cos\theta d\theta\)
\(\displaystyle =\frac{2a\sin\alpha}{3\alpha}\)
よって、重心の座標は\(\displaystyle \left(\frac{2a\sin\alpha}{3\alpha},0\right)\)

\(D\)の面積は
\(\displaystyle |D|=\int_0^1(1-\sqrt{x})^2dx=\frac{1}{6}\)
\(\displaystyle x_G=\frac{1}{|D|}\iint_Dxdxdy\)
\(\displaystyle =6\int_0^1\int_0^{(1-\sqrt{x})^2}xdydx\)
\(\displaystyle =6\int_0^1(x-2x^\frac{3}{2}+x^2)dx\)
\(\displaystyle =\frac{1}{5}\)
図形\(D\)は\(y=x\)に関して対称なので\(\displaystyle y_G=\frac{1}{5}\)
よって、重心の座標は\(\displaystyle \left(\frac{1}{5},\frac{1}{5}\right)\)

\(D\)の体積は
\(\displaystyle |D|=\frac{2\pi a^3}{3}\)
図形の対称性より\(x_G=y_G=0\)
\(\displaystyle z_G=\frac{1}{|D|}\iiint_Dzdxdydz\)
\(\displaystyle =\frac{3}{2\pi a^3}\int_0^a\iint_{D_z}zdxdydz\)
\(\displaystyle =\frac{3}{2\pi a^3}\int_0^a\pi z(a^2-z^2)dz\)
\(\displaystyle =\frac{3a}{8}\)
よって、重心の座標は\(\displaystyle \left(0,0,\frac{3a}{8}\right)\)

\(D\)の底面の中心を原点とし、頂点が\((0,0,h)\)とする。
\(D\)の体積は
\(\displaystyle |D|=\frac{\pi a^2h}{3}\)
図形の対称性より\(x_G=y_G=0\)
\(\displaystyle z_G=\frac{1}{|D|}\iiint_Dzdxdydz\)
\(\displaystyle =\frac{3}{\pi a^2h}\int_0^h\iint_{D_z}zdxdydz\)
\(\displaystyle =\frac{3}{\pi a^2h}\int_0^h\pi z\frac{(h-z)^2a^2}{h^2}dz\)
\(\displaystyle =\frac{h}{4}\)
よって、重心の座標は\(\displaystyle \left(0,0,\frac{h}{4}\right)\)