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22sp09-ex2-practice3-solutions

22sp09-ex2-practice3-solutions - a ØhÔ Ö iÒg9ÖaØ ieE...

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Unformatted text preview: a ØhÔ Ö iÒg9ÖaØ ieE ÜaÑ# Ú e Ö× iÓÒ a Öh99 h eÔ ÖÓb ÐeÑ ×ÓÒ Øh i×eÜaÑÛ e ÖeØak eÒ fÖÓÑ ea Ö Ðie ÖeÜaÑ ×ÓÒ Øh e×aÑ eØÓÔ iaÒd fÖÓÑ hÓÑ eÛ Ó Ök Ô ÖÓb ÐeÑ ×Ø i× iÒ ØeÒd ed ØÓg iÚ eÝÓÙaÒ id eaÓ fØh ed iÙ ÐØÝ ÐeÚ e Ð×ÓÔ eaÒd ÖaÒgeÓ fÔ ÖÓb ÐeÑ × Øha ØÝÓÙ ×hÓÙ Ðd eÜÔ eØØÓ×eeÓÒÓÙ Ö×eÓÒd4 :eÜaÑ ÓÛ eÚ e ÖiØdÓe×ÒÓ ØÓÚ e Öa ÐÐØÓÔ i×ÓÒ Øh e×Ý ÐÐabÙ × fÓ ÖØh i×eÜaÑY ÓÙ ×hÓÙ Ðd ÒÓ Ø a ××ÙÑ e ifag iÚ eÒ ØÝÔ eÓ fÔ ÖÓb ÐeÑ i×ÒÓ ØÓÒ Øh i×Ó ÖaÒÝ Ó Øh e ÖÔ ÖaØ ieeÜaÑ Øha Ø iØÛ ÓÙ ÐdÒÓ Øb eÓÒÝÓÙ ÖeÜaÑh e×Ý ÐÐabÙ ×fÓ ÖØh eeÜaÑ i××eØ iÓÒ × : 88995aÒd 5 Ó fÝÓÙ ÖØeÜ ØbÓÓk F iÒd Øh e ÐeÒg ØhÓ fØh eÙ ÖÚ e y = ln(cos( x )) x ∈ [0 ,π/ 4] ÔÓ iÒ Ø× Ó ÐÙ Ø iÓÒ Length = integraldisplay π/ 4 radicalBigg 1 + parenleftbigg- sin x cos x parenrightbigg 2 dx = integraldisplay π/ 4 radicalbig 1 + tan 2 xdx = integraldisplay π/ 4 sec xdx = ln | sec x + tan x || π/ 4 = ln( √ 2 + 1) . F iÒd Øh ea ÖeaÓ fØh e×Ù Ö faeÓ fÖeÚÓ ÐÙ Ø iÓÒÓb Øa iÒ edbÝ ÖÓ Øa Ø iÒg Øh eÙ ÖÚ e y = sin x x ∈ [0 ,π ] a ÖÓÙÒd Øh e x aÜ i× Ó ÐÙ Ø iÓÒ iÒeØh e×Ù Ö faea ÖeaÓ fag ÖaÔh ÖeÚÓ ÐÚ eda ÖÓÙÒdaÒaÜ i× i× integraltext 2 πrds Øh eÒ SA = integraldisplay 2 πrds = integraldisplay π 2 πy radicalBigg 1 + parenleftbigg dy dx parenrightbigg 2 dx = integraldisplay π 2 π sin( x ) radicalBig 1 + (cos( x )) 2 dx ×e Ø u = cos( x ) , =- integraldisplay − 1 1 2 π radicalBig 1 + ( u ) 2 du ÒÓÛ ×e Ø u = tan( θ ) =- integraldisplay − π 4 π 4 2 π sec( θ ) sec 2 ( θ ) dθ = integraldisplay π 4 − π 4 2 π sec 3 ( θ ) dθ = 2 π bracketleftBigg sec( θ ) tan( θ ) | π 4 − π 4- integraldisplay π 4 − π 4 tan 2 ( θ ) sec( θ ) dθ bracketrightBigg = 2 π bracketleftBigg sec( θ ) tan( θ ) | π 4 − π 4- integraldisplay π 4 − π 4 ( sec 2 ( θ )- 1 ) sec( θ ) dθ bracketrightBigg = 2 π bracketleftBigg sec( θ ) tan( θ ) | π 4 − π 4- integraldisplay π 4 − π 4 sec 3 ( θ ) dθ + integraldisplay π 4 − π 4 sec( θ ) dθ bracketrightBigg . ÓÛÑ ÓÚ eØh e integraltext π 4 − π 4 sec 3 ( θ ) dθ fÖÓÑ Øh e Ða ×Ø ÐiÒ eØÓ Øh eÚ e Ö× iÓÒ ÐiÒ e×ÙÔaÒdd iÚ id e bÝ 2 π integraldisplay π 4 − π 4 sec 3 ( θ ) dθ = 2 π bracketleftBigg sec( θ ) tan( θ ) | π 4 − π 4- integraldisplay π 4 − π 4 sec 3 ( θ ) dθ + integraldisplay π 4 − π 4 sec( θ ) dθ bracketrightBigg 4 π integraldisplay π 4 − π 4 sec 3 ( θ ) dθ = 2 π bracketleftBigg sec( θ ) tan( θ ) | π 4 − π 4 + integraldisplay π 4 − π 4 sec( θ ) dθ bracketrightBigg 2 π integraldisplay...
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22sp09-ex2-practice3-solutions - a ØhÔ Ö iÒg9ÖaØ ieE...

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