1. —‡•–‹‘ ‘—” ǥ
ܑ െ මሺ૜‫ ܠ‬൅ ૛‫ܡ‬ሻ‫ ܢ܌ܡ܌ܠ܌‬ǥ
‫܁‬
Where S is the solid bounded above by the plane z=4, below by the plane z=0 and laterally by
the cylinder with generators parallel to the z-axis over the boundary of the rectangular region
ǣ െ ͳ ൑ š ൑ ͳǡ െ ͳ ൑ › ൑ ͵Ǥ
The Answer …
š ൌ െͳ –‘ ͳ › ൌ െͳ –‘ ͵ œ ൌ Ͳ –‘ Ͷ
ଵ ଷ ସ ଵ ଷ ସ
න න න ሺ͵š ൅ ʹ›ሻ†œ†›†š ൌ න න ቈන ሺ͵š ൅ ʹ›ሻ†œ቉ †›†š
ିଵ ିଵ ଴ ିଵ ିଵ ଴
͵šœ ʹ›œ ସ
ଵ ଷ ଵ ଷ ଵ ଷ
ൌන න ൤ ൅ ൨ †›†š ൌ න න ሾͳʹš ൅ ͺ›ሿ†›†š ൌ න ቈන ሾͳʹš ൅ ͺ›ሿ†›቉ †š
ିଵ ିଵ ͳ ͳ ଴ ିଵ ିଵ ିଵ ିଵ
ଷ ଵ
ͺ› ଶ
ଵ ଵ
Ͷͺš ଶ
ൌ න ቈͳʹš› ൅ ቉ †š ൌ න ሾͶͺš ൅ ͵ʹሿ†š ൌ ቈ ൅ ͵ʹš቉
ିଵ ʹ ିଵ ିଵ ʹ ିଵ
ሾʹͶሺͳሻଶ ൅ ͵ʹሺͳሻሿ െ ሾʹͶሺെͳሻଶ ൅ ͵ʹሺെͳሻሿ ൌ ͷ͸ ൅ ͺ ൌ ૟૝ ͓
ܑܑ െ මሺ‫ ܠ‬૛ ሻ‫ ܢ܌ܡ܌ܠ܌‬ǥ ‫ ܠ ׷ ܁ ܍ܚ܍ܐ܅‬૛ ൅ ‫ ܡ‬૛ ൑ ૝ ‫ ܌ܖ܉‬૙ ൑ ‫ ܢ‬൑ ૞Ǥ
‫܁‬
The Answer …
‘˜‡”– –Š‡ ‹–‡‰”ƒŽ –‘ ƒ …›Ž‹†”‹…ƒŽ …‘‘”†‹ƒ–‡•
† ൌ †š†› ൌ ”†œ†”†Ʌ
මሺš ଶ ሻ† Ʌ ൌ Ͳ –‘ ʹɎ ” ൌ Ͳ –‘ ξͶ œ ൌ Ͳ –‘ ͷ
ୗ
”‘ ‘Žƒ” ‘‘”†‹ƒ–‡•
š ൌ ” …‘•ሺɅሻ ƒ† › ൌ ” •‹ሺɅሻ
ଶ஠ ଶ ହ ଶ஠ ଶ ହ
‫׵‬න න න ሾ” …‘•ሺɅሻሿ ”†œ†”†Ʌ ൌ න ଶ
න ቈන ”ሾ” …‘•ሺɅሻሿଶ †œ቉ †”†Ʌ
଴ ଴ ଴ ଴ ଴ ଴
ହ
ଶ஠ ଶ ହ ଶ஠ ଶ
œ” ଷ …‘• ଶ ሺɅሻ
ൌන න ቈන ሾ” ଷ …‘• ሺɅሻሿ†œ቉ †”†Ʌ ൌ න
ଶ
න ቈ ቉ †”†Ʌ ՜
଴ ଴ ଴ ଴ ଴ ͳ ଴
ϲͮWĂŐĞ

2. ଶ஠ ଶ ଶ஠ ଶ
՜ ൌන න ሾͷ” ଷ …‘• ሺɅሻሿ†”†Ʌ ൌ න
ଶ
ቈන ሾͷ” ଷ …‘• ଶ ሺɅሻሿ†”቉ †Ʌ
଴ ଴ ଴ ଴
ଶ
ଶ஠
ͷ” ସ …‘• ଶ ሺɅሻ ଶ஠ ଶ஠
ͳ
ൌන ቈ ቉ †Ʌ ൌ න ሾʹͲ …‘•ଶ ሺɅሻሿ†Ʌ ൌ ʹͲ න ൤ …‘•ሺʹɅ ൅ ͳሻ൨ †Ʌ
଴ Ͷ ଴ ଴ ଴ ʹ
ଶ஠
ଶ஠
•‹ሺʹɅ ൅ ͳሻ ଶ஠ •‹ሺɅሻ …‘•ሺɅሻ ൅ Ʌ
ൌ ͳͲ න ሾ…‘•ሺʹɅ ൅ ͳሻሿ†Ʌ ൌ ͳͲ ൤ ൨ ൌ ͳͲ ቈ ቉
଴ ʹ ଴ ͳ ଴
ൌ ͳͲሺʹɎሻ ൌ ૛૙ૈ ͓
ܑܑܑ െ මሺ૜‫ܡܠ‬ሻ‫ ܢ܌ܡ܌ܠ܌‬ǥ
‫܁‬
Where S is the solid in the first octant bounded by the coordinate planes and the plane
š ൅ › ൅ œ ൌ ͸Ǥ
The Answer …
”‘Œ‡…–‹‰ ‘–‘ –Š‡ šǦ› ’Žƒ‡ǣ ‡‰‹‘ „‡–™‡‡ šαͲǡ ›αͲǡ ƒ† šΪ›α͸
œൌͲ š൅›ൌ͸ ՜ ›ൌ͸െš ‫ › ׵‬ൌ Ͳ –‘ ͸ െ š
œ ൌ Ͳǡ › ൌ Ͳ ƒ† š ൅ › ൌ ͸ ‫ š ׵‬ൌ Ͳ –‘ ͸
š൅›൅œൌ ͸ ՜ œൌ͸െšെ› ‫ œ ׵‬ൌ Ͳ –‘ ͸ െ š െ ›
଺ ଺ି୶ ଺ି୶ି୷ ଺ ଺ି୶ ଺ି୶ି୷
‫׵‬න න න ሺ͵š›ሻ†œ†›†š ൌ න න ቈන ሺ͵š›ሻ†œ቉ †›†š
଴ ଴ ଴ ଴ ଴ ଴
଺ ଺ି୶
͵š›œ ଺ି୶ି୷ ଺ ଺ି୶
ൌන න ൤ ൨ †›†š ൌ න න ሾͳͺš› െ ͵š ଶ › െ ͵š› ଶ ሿ†›†š
଴ ଴ ͳ ଴ ଴ ଴
଺ି୶
଺ ଺ି୶ ଺
͵
ൌ න ቈන ሾͳͺš› െ ͵š ଶ › െ ͵š› ଶ ሿ†›቉ †š ൌ න ൤ͻš› ଶ െ š ଶ › ଶ െ š› ଷ ൨ †š
଴ ଴ ଴ ʹ ଴
଺
െͳ െͳ ଺ ସ
ൌන ൤ ሺš െ ͸ሻଷ š൨ †š ൌ න ሾš െ ͳͺš ଷ ൅ ͳͲͺš ଶ െ ʹͳ͸šሿ†š
଴ ʹ ʹ ଴
଺
െͳ š ହ ͻš ସ െͳ ૢૠ૛
ൌ ቈ െ ൅ ͵͸š ଷ െ ͳͲͺš ଶ ቉ ൌ ሺെ͵ͺͺǤͺሻ ൌ ͓
ʹ ͷ ʹ ଴
ʹ ૞
ǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦǦ
ϳͮWĂŐĞ