Exercise \(\PageIndex{3}\)
In the following exercises, consider a lamina occupying the region \(R\) and having the density function \(\rho\) given in the first two groups of Exercises.
a. Find the moments of inertia \(I_x, \space I_y\) and \(I_0\) about the \(x\)-axis, \(y\)-axis, and origin, respectively.
b. Find the radii of gyration with respect to the \(x\)-axis, \(y\)-axis, and origin, respectively.
1. \(R\) is the triangular region with vertices \((0,0), \space (0,3)\), and \((6,0); \space \rho (x,y) = xy\).
- Answer
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a. \(I_x = \frac{243}{10}, \space I_y = \frac{486}{5}\), and \(I_0 = \frac{243}{2}\); b. \(R_x = \frac{3\sqrt{5}}{5}, \space R_y = \frac{6\sqrt{5}}{5}\), and \(R_0 = 3\)
2. \(R\) is the triangular region with vertices \((0,0), \space (1,1)\), and \((0,5); \space \rho (x,y) = x + y\).
3. \(R\) is the rectangular region with vertices \((0,0), \space (0,3), \space (6,3)\), and \((6,0); \space \rho (x,y) = \sqrt{xy}\).
- Answer
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a. \(I_x = \frac{2592\sqrt{2}}{7}, \space I_y = \frac{648\sqrt{2}}{7}\), and \(I_0 = \frac{3240\sqrt{2}}{7}\); b. \(R_x = \frac{6\sqrt{21}}{7}, \space R_y = \frac{3\sqrt{21}}{7}\), and \(R_0 = \frac{3\sqrt{106}}{7}\)
4. \(R\) is the rectangular region with vertices \((0,1), \space (0,3), \space (3,3)\), and \((3,1); \space \rho (x,y) = x^2y\).
5. \(R\) is the trapezoidal region determined by the lines \(y = - \frac{1}{4}x + \frac{5}{2}, \space y = 0, \space y = 2\), and x = 0; \space \rho (x,y) = 3xy\).
- Answer
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a. \(I_x = 88, \space I_y = 1560\), and \(I_0 = 1648\); b. \(R_x = \frac{\sqrt{418}}{19}, \space R_y = \frac{\sqrt{7410}}{10}\), and \(R_0 = \frac{2\sqrt{1957}}{19}\)
6. \(R\) is the trapezoidal region determined by the lines \(y = 0, \space y = 1, \space y = x\), and y = -x + 3; \space \rho (x,y) = 2x + y\).
7. \(R\) is the disk of radius \(2\) centered at \((1,2); \space \rho(x,y) = x^2 + y^2 - 2x - 4y + 5\).
- Answer
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a. \(I_x = \frac{128\pi}{3}, \space I_y = \frac{56\pi}{3}\), and \(I_0 = \frac{184\pi}{3}\); b. \(R_x = \frac{4\sqrt{3}}{3}, \space R_y = \frac{\sqrt{21}}{2}\), and \(R_0 = \frac{\sqrt{69}}{3}\)
8. \(R\) is the unit disk; \(\rho (x,y) = 3x^4 + 6x^2y^2 + 3y^4\).
9. \(R\) is the region enclosed by the ellipse \(x^2 + 4y^2 = 1; \space \rho(x,y) = 1\).
- Answer
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a. \(I_x = \frac{\pi}{32}, \space I_y = \frac{\pi}{8}\), and \(I_0 = \frac{5\pi}{32}\); b. \(R_x = \frac{1}{4}, \space R_y = \frac{1}{2}\), and \(R_0 = \frac{\sqrt{5}}{4}\)
10. \(R = \{(x,y) | 9x^2 + y^2 \leq 1, \space x \geq 0, \space y \geq 0\} ; \space \rho (x,y) = \sqrt{9x^2 + y^2}\).
11. \(R\) is the region bounded by \(y = x, \space y = -x, \space y = x + 2\), and \(y = -x + 2; \space \rho (x,y) = 1\).
- Answer
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a. \(I_x = \frac{7}{3}, \space I_y = \frac{1}{3}\), and \(I_0 = \frac{8}{3}\); b. \(R_x = \frac{\sqrt{42}}{6}, \space R_y = \frac{\sqrt{6}}{6}\), and \(R_0 = \frac{2\sqrt{3}}{3}\)
12. \(R\) is the region bounded by \(y = \frac{1}{x}, \space y = \frac{2}{x}, \space y = 1\), and \(y = 2; \space \rho (x,y) = 4(x + y)\).