Gerald B. Folland's Advanced Calculus (Solutions Manual) PDF

By Gerald B. Folland

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8. See answer in back of text. 9. (a) The region is bounded below by ⑧the ➇❢of ✄✳integration ✏❝✍✪➁ ✟ ✄❈➁ region in the first quadrant of the the parabola and above by the plane . (b) and (c): See answers in back of text. ✠ ☎ €✠ ✠ ✝ ✞ of mass is ✪ ✱ ➇❍➁ -plane under ☎ ✠ ✌ ✌ ✆▼☞✎✏ ✍ ☎✠ ✍ ✠ ✗ ✒ ✌ ➁ ✌ ➇❈✄ ✠ ✟€ ✆▼☞✎✏✑✍ ☎✠ ✒ ✟ ✌ ➇❈✄ ✓€ ✆ . The ➇ -moment is ☎✠ ✍ ✗ ✒ ✌ ➁ ✌ ➇⑥✄ ✟€ ✆ ➇■☞✎✏❑✍ ☎✠ ✒ ✟ ✌ € ➇★✄ ✟ € € ✟ ✆ , so the ➇ -coordinate of the center ➁ ✆ ☞€ ✎ ☞ 10.

A) Since the first is contained in the square ❷ , using polar coordinates we have ✤ ✆ ✒ ✟quadrant ✍⑩✼✙✔ disc ✕ ✟ ✁ ✁ ✁ ✎ ✔✸✻✺✜✼ ✟ could also be done by a € ✝ ✝ ❵ ☛✟ ✝ ✝ ✵ ✵ ✾✵ ; the -integral diverges. ) calculation like that in part (b) below,€ using the fact that ✝ ✝ ❵ ☛✟ ✝ ✝ ❵ ✁ ✌ ✌ ✌ ✁ ✌ ☎ ✌ ✌ ☞❊➇❄➂❨➁❬✒ ✄ ☞❊➇ ✟ ✡✌➁ ✟ ✒ ✎ ❈ ✍ ✙▲➁ ✟ ☞❊➇ ✟ ✡✕➁ ✟ ✒ ✎ ✟ ➁ ✙ ✰ ➇✚✄ € ✸✻✺✜✼❾✙ (b) have ❵ ☞❊➇❄➂❨➁❬✒ ➇ ➁ ✂❜✶ We ✕ ➁ ✎ € , so☞✎✏ for ✍⑤✙■a ✸✻fixed ✺✜✼ ✟ ✒ ✄ the ✍ ➁✢✎ substitution ✯ ✄ € € € ✝ ✝ into ✁ turns the indefinite integral ✝ ❵ ✌ ✁ ✎✁ ✌ ✁ ✌✔✁ € ☞❊➇❄➂❨➁❬✒ ➇★✄✳✍✑✏✓✽✣☞✎✏❑✡ ➁ ✟ ✒ ☞❊➇❄➂❨➁❬✒ ➇ ➁✉✄ ✍✯➁ ✎ ✼✙✔ ✕ ✸✻✺✜✼ ✄✳✍❝➇❴✽✣☞❊➇ ✟ ✡ ➁ ✟ ✒ ✁ ✁ .

Have ✂ . Hence the arc length is ✸✻✺✜✼ ✟❴❛ ✂ ✆ ✒ ✟ ✏❑✡ ✸✻✺✜✼ ✟➃❛ ❛ ✄ ✼✙✔ ✕ ✟ ❛❫✡❲✸✻✺✜✼ ✟ ❛❫✡ ☞ ✙✑✍⑩ ❁ ❛ ✄ ✙■✼✄✔ ✕❤❛❜✒ ✌ ☞ ✙✑✍⑩✙■✼✄✔ ✕❤❛❜✒ ✌ ✒ ✒ ✎ ✆ ✟ ✎ ✆ ✟ ✒ ✒ ✆ ✟ ✧✗✧✗✧ ❛❝✄✳✙ ✆ ✟ ✧✗✧✗✧ ✂ ✙ ◗ ✂✲✄ € ☞ € ❃❦✍⑩❛❨✒ ❛❤✄ ✟€ ❃❦✍◗✙✄✂ ✍ ✙■✼✄✔ ✕✯❛✯✄✳✙ ✍◗✙■✸✻✺✜✼❾✙✄✂⑧✄ ✒ ✟ ✟ Let , so . Then ✝ ✎ ✆ ✟ ; ✌ ✝ ✌ ❁✼✄✔ ✕ ✟ ✂ ✸✻✺✜✼ ✟ ❛ ✄ ✄✼ ✔ ✕ ✟ ✄✙ ✂ ✄ ❁✄✼ ✔ ✕ ✟ ✂✬✸✻✺✜✼ ✟ ✂ ✞ ✄ ✙ ✆ ✒ ✟ ✛ ✏❑✡❲✸✻✺✜✼ ✟ ✂ ✂ ✄ , and , so ✝ ✌ ✙ ✆ ✒ ✟ ✆ ✙✘✍✪✼✄✔✖✕ ✟ ✂ ✂✦✄✕✙ ✜ ✒ ✟ ☞ ✙ ✎ ✓€ ✒ ✟ ✒ ✞◗✄ ✒ ✆ ✟ ✌ ✝ ✄ ✟ ✝ ✎ 4. 5. ✌ ✂✑✄ ✙■✼ ✆ ✁ ✘✏❑✡❲✏ ✼ ✟ ➇ ➇❋✄⑨✸✻✺✜✼✄ ❝➇ ➇ ✌ ➁ ✌ .

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Advanced Calculus (Solutions Manual) by Gerald B. Folland


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