Physical Applications

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

Moments and Centers of Mass

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