轉動慣量計算器
逐步求解區域/面積的截面二次矩與回轉半徑
此計算器將嘗試求出由給定曲線所圍區域的面積慣性矩與回轉半徑，並顯示步驟。
您的輸入
求由曲線 $$$y = 3 x$$$, $$$y = x^{2}$$$ 所圍成之區域的慣性矩。
解答
$$$I_{x} = \int\limits_{0}^{3}\int\limits_{x^{2}}^{3 x} y^{2}\cdot 1\, dy\, dx = \frac{2187}{28}\approx 78.107142857142857$$$
$$$I_{y} = \int\limits_{0}^{3}\int\limits_{x^{2}}^{3 x} x^{2}\cdot 1\, dy\, dx = \frac{243}{20} = 12.15$$$
$$$m = \int\limits_{0}^{3}\int\limits_{x^{2}}^{3 x} 1\, dy\, dx = \frac{9}{2} = 4.5$$$
$$$R_{x} = \sqrt{\frac{I_{x}}{m}} = \frac{9 \sqrt{42}}{14}\approx 4.166190448976482$$$
$$$R_{y} = \sqrt{\frac{I_{y}}{m}} = \frac{3 \sqrt{30}}{10}\approx 1.643167672515498$$$
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