Physical Applications

1. [latex]150[/latex] ft-lb
2. [latex]200\text{J}[/latex]
3. [latex]1[/latex] J
4. [latex]\frac{39}{2}[/latex]
5. [latex]\text{ln}(243)[/latex]
6. [latex]\frac{332\pi }{15}[/latex]
7. [latex]100\pi[/latex]
8. [latex]20\pi \sqrt{15}[/latex]
9. [latex]6[/latex] J
10. [latex]5[/latex] cm
11. [latex]36[/latex] J
12. [latex]18,750[/latex] ft-lb
13. [latex]\frac{32}{3}×{10}^{9}\text{ft-lb}[/latex]
14. [latex]8.65×{10}^{5}\text{J}[/latex]
15. 1. [latex]3,000,000[/latex] lb,
    2. [latex]749,000[/latex] lb
16. [latex]23.25\pi[/latex] million ft-lb
17. [latex]\frac{A\rho {H}^{2}}{2}[/latex]
18. Answers may vary

Moments and Centers of Mass

1. [latex]\frac{5}{4}[/latex]
2. [latex](\frac{2}{3},\frac{2}{3})[/latex]
3. [latex](\frac{7}{4},\frac{3}{2})[/latex]
4. [latex]\frac{3L}{4}[/latex]
5. [latex]\frac{\pi }{2}[/latex]
6. [latex]\frac{{e}^{2}+1}{{e}^{2}-1}[/latex]
7. [latex]\frac{{\pi }^{2}-4}{\pi }[/latex]
8. [latex]\frac{1}{4}(1+{e}^{2})[/latex]
9. [latex](\frac{a}{3},\frac{b}{3})[/latex]
10. [latex](0,\frac{\pi }{8})[/latex]
11. [latex](0,3)[/latex]
12. [latex](0,\frac{4}{\pi })[/latex]
13. [latex](\frac{5}{8},\frac{1}{3})[/latex]
14. [latex]\frac{m\pi }{3}[/latex]
15. [latex]\pi {a}^{2}b[/latex]
16. [latex](\frac{4}{3\pi },\frac{4}{3\pi })[/latex]
17. [latex](\frac{1}{2},\frac{2}{5})[/latex]
18. [latex](0,\frac{28}{9\pi })[/latex]
19. Center of mass: [latex](\frac{a}{6},\frac{4{a}^{2}}{5}),[/latex] volume: [latex]\frac{2\pi {a}^{4}}{9}[/latex]
20. Volume: [latex]2{\pi }^{2}{a}^{2}(b+a)[/latex]