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

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

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

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

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

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

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 \(y = 2; \space \rho (x,y) = 4(x + y)\).