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Lekcja 6 – Zastosowania całki oznaczonej – krzywe w postaci biegunowej

Jesteś tutaj: Strona główna / Fora / Lekcja 6 – Postać biegunowa (VIDEO)

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  • Ten temat ma 2 odpowiedzi, 1 udzielający się, ostatnio wpisał/a coś 1 rok, 8 miesiące temu Krystian Karczyński.
  • Autor
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  • #13726
    Krystian Karczyński
    Dyrektor

      Pobierz wzory na zastosowania całek (PDF) Pobierz wzory na całki (PDF) Pobierz tablice trygonometryczne (PDF) Pobierz schemat na całki wymierne
    [Zobacz cały post na stronie: Lekcja 6 – Postać biegunowa (VIDEO)]

    #13345
    Krystian Karczyński
    Dyrektor

    Pobierz Zadanie Domowe (PDF) Pobierz Rozwiązanie Zadania Domowego (PDF)
    [Zobacz cały post na stronie: Zadanie domowe do Lekcji 6]

    #40992
    Krystian Karczyński
    Dyrektor

    Zadanie 7

    Oblicz pola figur ograniczonych krzywymi:

    x to the power of 4 plus y to the power of 4 equals 25 open parentheses x squared plus y squared close parentheses

     

    Dla ciekawych, wykres tej krzywej wyglądał by tak:

    open curly brackets table attributes columnalign left end attributes row cell x equals \rho cos phi end cell row cell y equals \rho sin phi end cell end table close

    open parentheses \rho cos phi close parentheses to the power of 4 plus open parentheses \rho sin phi close parentheses to the power of 4 equals 25 open square brackets open parentheses \rho cos phi close parentheses squared plus open parentheses \rho sin phi close parentheses squared close square brackets
\rho to the power of 4 cos to the power of 4 phi plus \rho to the power of 4 sin to the power of 4 phi equals 25 open parentheses \rho squared cos squared phi plus \rho squared sin squared phi close parentheses
\rho to the power of 4 open parentheses cos to the power of 4 phi plus sin to the power of 4 phi close parentheses equals 25 \rho squared open parentheses cos squared phi plus sin squared phi close parentheses
\rho squared open parentheses cos to the power of 4 phi plus sin to the power of 4 phi close parentheses equals 25
\rho squared equals fraction numerator 25 over denominator cos to the power of 4 phi plus sin to the power of 4 phi end fraction
\rho equals fraction numerator 5 over denominator square root of cos to the power of 4 phi plus sin to the power of 4 phi end root end fraction

    \rho equals fraction numerator 5 over denominator square root of cos to the power of 4 phi plus 2 cos squared phi sin squared phi plus sin to the power of 4 phi minus 2 cos squared phi sin squared phi end root end fraction
\rho equals fraction numerator 5 over denominator square root of open parentheses cos squared phi plus sin squared phi close parentheses squared minus 1 half times 4 cos squared phi sin squared phi end root end fraction
\rho equals fraction numerator 5 over denominator square root of 1 minus 1 half times open parentheses 2 cos phi sin phi close parentheses squared end root end fraction
\rho equals fraction numerator 5 over denominator square root of 1 minus 1 half sin squared 2 phi end root end fraction

    Teraz, aby dalej przekształcić sin squared 2 phi wyciągam wzór na cosinus podwojonego kąta:

    cos 4 phi equals cos squared 2 phi minus sin squared 2 phi
cos 4 phi equals 1 minus sin squared 2 phi minus sin squared 2 phi
cos 4 phi equals 1 minus 2 sin squared 2 phi
2 sin squared 2 phi equals 1 minus cos 4 phi
sin squared 2 phi equals 1 half open parentheses 1 minus cos 4 phi close parentheses

    Wracam do wzoru na \rho:

    \rho equals fraction numerator 5 over denominator square root of 1 minus 1 half times 1 half open parentheses 1 minus cos 4 phi close parentheses end root end fraction
\rho equals fraction numerator 5 over denominator square root of 1 minus 1 fourth open parentheses 1 minus cos 4 phi close parentheses end root end fraction
\rho equals fraction numerator 5 over denominator square root of 1 minus 1 fourth plus 1 fourth cos 4 phi end root end fraction
\rho equals fraction numerator 5 over denominator square root of 3 over 4 plus 1 fourth cos 4 phi end root end fraction
\rho equals fraction numerator 5 over denominator square root of 1 fourth open parentheses 3 plus cos 4 phi close parentheses end root end fraction
\rho equals fraction numerator 5 over denominator square root of 1 fourth end root square root of 3 plus cos 4 phi end root end fraction
\rho equals fraction numerator 5 over denominator 1 half square root of 3 plus cos 4 phi end root end fraction
\rho equals fraction numerator 10 over denominator square root of 3 plus cos 4 phi end root end fraction

    Jak widać, wartość pod pierwiastkiem jest zawsze dodatnia, a także mianownik nie jest nigdy równy 0. Możemy więc przyjąć, że mamy phi element of open \angle brackets 0 comma 2 \pi close \angle brackets.

    Podstawiamy do wzoru na pole obszaru dla krzywej w postaci biegunowej:

    P equals 1 half integral subscript 0 superscript 2 \pi end superscript open parentheses fraction numerator 10 over denominator square root of 3 plus cos 4 phi end root end fraction close parentheses squared d phi equals 1 half integral subscript 0 superscript 2 \pi end superscript fraction numerator 100 over denominator 3 plus cos 4 phi end fraction d phi equals...

    Liczymy całkę nieoznaczoną na boku podstawieniem uniwersalnym:

    integral fraction numerator 100 over denominator 3 plus cos 4 phi end fraction d phi equals open vertical bar table row cell t equals 4 phi end cell row cell d t equals 4 d phi end cell row cell fraction numerator d t over denominator 4 end fraction equals d phi end cell end table close vertical bar equals integral fraction numerator 100 over denominator 3 plus cos t end fraction fraction numerator d t over denominator 4 end fraction equals integral fraction numerator 25 over denominator 3 plus cos t end fraction d t equals
equals open vertical bar table row cell u equals t g begin inline style t over 2 end style end cell row cell cos t equals fraction numerator 1 minus u squared over denominator 1 plus u squared end fraction end cell row cell d t equals fraction numerator 2 d u over denominator 1 plus u squared end fraction end cell end table close vertical bar equals integral fraction numerator 25 over denominator 3 plus fraction numerator 1 minus u squared over denominator 1 plus u squared end fraction end fraction fraction numerator 2 d u over denominator 1 plus u squared end fraction equals integral fraction numerator 50 over denominator 3 open parentheses 1 plus u squared close parentheses plus 1 minus u squared end fraction d u equals
equals integral fraction numerator 50 over denominator 4 plus 2 u squared end fraction d u equals integral fraction numerator 50 over denominator 2 open parentheses 2 plus u squared close parentheses end fraction d u equals 25 integral fraction numerator 1 over denominator open parentheses square root of 2 close parentheses squared plus u squared end fraction d u equals
equals 25 fraction numerator 1 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator u over denominator square root of 2 end fraction end style plus C equals fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g t over 2 over denominator square root of 2 end fraction end style plus C equals fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g fraction numerator 4 phi over denominator 2 end fraction over denominator square root of 2 end fraction end style plus C equals
equals fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g 2 phi over denominator square root of 2 end fraction end style plus C

     

    Jadąc dalej napotykamy jednak spore kłopoty i dosyć rzadko spotykany w praktyce przypadek..

    Schematycznie, bierzemy wynik z całki nieoznaczonej i wracamy się do całki oznaczonej:

    P equals 1 half integral subscript 0 superscript 2 \pi end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi equals 1 half right enclose open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g 2 phi over denominator square root of 2 end fraction end style close square brackets end enclose subscript 0 superscript 2 \pi end superscript equals
equals 1 half open parentheses open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g open parentheses 2 times 2 \pi close parentheses over denominator square root of 2 end fraction end style close square brackets minus open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g open parentheses 2 times 0 close parentheses over denominator square root of 2 end fraction end style close square brackets close parentheses equals
equals 1 half open parentheses open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g 0 over denominator square root of 2 end fraction end style close square brackets minus open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g 0 over denominator square root of 2 end fraction end style close square brackets close parentheses equals 1 half open parentheses fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style 0 end style minus fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style 0 end style close parentheses equals
equals 1 half times 0 equals 0

    Wynik jest zaskakujący i na pewno nieprawidłowy, bo przecież liczymy całkę z funkcji nieujemnej fraction numerator 100 over denominator 3 plus 4 cos phi end fraction , wynik nie może wyjść zero.

    Bardzo przepraszam, w Kursie nie podkreśliłem wystarczająco mocno pewne założenie, na którym opiera się cały schemat. Mianowicie podstawowy wzór całkowy na całkę oznaczoną, czyli:

    integral subscript a superscript b f open parentheses x close parentheses d x equals F open parentheses b close parentheses minus F open parentheses a close parentheses
g d z i e colon
integral f open parentheses x close parentheses d x equals F open parentheses x close parentheses plus C

    można stosować tylko wtedy, gdy funkcja F open parentheses x close parentheses jest ciągła i różniczkowalna w przedziale open \angle brackets a comma b close \angle brackets.

    Przyjrzyjmy się naszej funkcji F open parentheses x close parentheses w naszym konkretnym przykładzie.

    fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g 2 phi over denominator square root of 2 end fraction end style w przedziale phi element of open \angle brackets 0 comma 2 \pi close \angle brackets

    Funkcja tangens nie jest ciągła dla argumentów \pi over 2 comma 3 over 2 \pi comma 5 over 2 \pi comma 7 over 2 pi, czyli nasza F open parentheses x close parentheses nie jest ciągła dla argumentów:

    2 phi equals \pi over 2 space logical or space 2 phi equals 3 over 2 \pi space logical or space 2 phi equals 5 over 2 \pi space logical or space 2 phi equals 7 over 2 pi
phi equals \pi over 4 space logical or space phi equals 3 over 4 \pi space logical or space phi equals 5 over 4 \pi space logical or space phi equals 7 over 4 pi

    Aby zastosować podstawowy wzór całkowy integral subscript a superscript b f open parentheses x close parentheses d x equals F open parentheses b close parentheses minus F open parentheses a close parentheses należy więc porozbijać naszą całkę na całki niewłaściwe:

    P equals 1 half integral subscript 0 superscript 2 \pi end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi equals 1 half left parenthesis integral subscript 0 superscript begin inline style \pi over 4 end style end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi plus integral subscript begin inline style \pi over 4 end style end subscript superscript begin inline style \pi over 2 end style end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi plus
plus integral subscript begin inline style \pi over 2 end style end subscript superscript begin inline style fraction numerator 3 \pi over denominator 4 end fraction end style end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi plus integral subscript begin inline style fraction numerator 3 \pi over denominator 4 end fraction end style end subscript superscript begin inline style straight \pi end style end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi plus integral subscript begin inline style straight \pi end style end subscript superscript begin inline style fraction numerator 5 \pi over denominator 4 end fraction end style end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi plus
plus integral subscript begin inline style fraction numerator 5 straight \pi over denominator 4 end fraction end style end subscript superscript begin inline style fraction numerator 3 \pi over denominator 2 end fraction end style end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi plus integral subscript begin inline style fraction numerator 3 \pi over denominator 2 end fraction end style end subscript superscript begin inline style fraction numerator 7 \pi over denominator 4 end fraction end style end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi plus integral subscript begin inline style fraction numerator 7 \pi over denominator 4 end fraction end style end subscript superscript begin inline style 2 straight \pi end style end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi right parenthesis

    Zacznijmy liczyć całki (całkę nieoznaczoną do każdej z nich mamy już policzoną, nie będzie tak źle ? ):

    integral subscript 0 superscript begin inline style \pi over 4 end style end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi equals limit as epsilon rightwards arrow begin inline style straight \pi over 4 end style to the power of minus of right enclose open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g 2 phi over denominator square root of 2 end fraction end style close square brackets end enclose subscript 0 superscript epsilon equals
equals limit as epsilon rightwards arrow begin inline style straight \pi over 4 to the power of minus end style of open parentheses open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g 2 epsilon over denominator square root of 2 end fraction end style close square brackets minus open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g open parentheses 2 times 0 close parentheses over denominator square root of 2 end fraction end style close square brackets close parentheses equals fraction numerator 25 straight \pi over denominator 2 square root of 2 end fraction minus 0 equals fraction numerator 25 straight \pi over denominator 2 square root of 2 end fraction
open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g 2 times straight \pi over 4 over denominator square root of 2 end fraction end style close square brackets equals open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g straight \pi over 2 over denominator square root of 2 end fraction end style close square brackets equals open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator plus infinity over denominator square root of 2 end fraction end style close square brackets equals
equals open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style open parentheses plus infinity close parentheses end style close square brackets equals open square brackets fraction numerator 25 over denominator square root of 2 end fraction times straight \pi over 2 close square brackets equals open square brackets fraction numerator 25 straight \pi over denominator 2 square root of 2 end fraction close square brackets

     

     

    Bierzemy drugą całkę:

    integral subscript begin inline style \pi over 4 to the power of plus end style end subscript superscript begin inline style \pi over 2 end style end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi equals limit as epsilon rightwards arrow begin inline style straight \pi over 4 end style to the power of plus of right enclose open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g 2 phi over denominator square root of 2 end fraction end style close square brackets end enclose subscript epsilon superscript begin inline style \pi over 2 end style end superscript equals
equals limit as epsilon rightwards arrow begin inline style straight \pi over 4 to the power of minus end style of open parentheses open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g open parentheses 2 times \pi over 2 close parentheses over denominator square root of 2 end fraction end style close square brackets minus open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g open parentheses 2 times epsilon close parentheses over denominator square root of 2 end fraction end style close square brackets close parentheses equals
equals limit as epsilon rightwards arrow begin inline style straight \pi over 4 to the power of minus end style of open parentheses open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g straight \pi over denominator square root of 2 end fraction end style close square brackets minus open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g open parentheses 2 epsilon close parentheses over denominator square root of 2 end fraction end style close square brackets close parentheses equals 0 minus open square brackets negative fraction numerator 25 straight \pi over denominator 2 square root of 2 end fraction close square brackets equals fraction numerator 25 straight \pi over denominator 2 square root of 2 end fraction
open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g 2 times straight \pi over 4 over denominator square root of 2 end fraction end style close square brackets equals open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator t g straight \pi over 2 over denominator square root of 2 end fraction end style close square brackets equals open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style fraction numerator negative infinity over denominator square root of 2 end fraction end style close square brackets equals
equals open square brackets fraction numerator 25 over denominator square root of 2 end fraction a r c t g begin inline style open parentheses negative infinity close parentheses end style close square brackets equals open square brackets fraction numerator 25 over denominator square root of 2 end fraction times open parentheses negative straight \pi over 2 close parentheses close square brackets equals open square brackets negative fraction numerator 25 straight \pi over denominator 2 square root of 2 end fraction close square brackets

    Jak widać, wyniki pierwszej i drugiej całki wyszły takie same. Całek do policzenia jest 8, ale z symetryczności funkcji tangens można wywnioskować, że wynik każdej całki będzie taki sam.

    Mam więc:

    P equals 1 half integral subscript 0 superscript 2 \pi end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi equals 1 half left parenthesis integral subscript 0 superscript begin inline style \pi over 4 end style end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi plus integral subscript begin inline style \pi over 4 end style end subscript superscript begin inline style \pi over 2 end style end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi plus
plus integral subscript begin inline style \pi over 2 end style end subscript superscript begin inline style fraction numerator 3 \pi over denominator 4 end fraction end style end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi plus integral subscript begin inline style fraction numerator 3 \pi over denominator 4 end fraction end style end subscript superscript begin inline style straight \pi end style end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi plus integral subscript begin inline style straight \pi end style end subscript superscript begin inline style fraction numerator 5 \pi over denominator 4 end fraction end style end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi plus
plus integral subscript begin inline style fraction numerator 5 straight \pi over denominator 4 end fraction end style end subscript superscript begin inline style fraction numerator 3 \pi over denominator 2 end fraction end style end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi plus integral subscript begin inline style fraction numerator 3 \pi over denominator 2 end fraction end style end subscript superscript begin inline style fraction numerator 7 \pi over denominator 4 end fraction end style end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi plus integral subscript begin inline style fraction numerator 7 \pi over denominator 4 end fraction end style end subscript superscript begin inline style 2 straight \pi end style end superscript fraction numerator 100 over denominator 3 plus 4 cos phi end fraction d phi right parenthesis equals
equals 1 half left parenthesis fraction numerator 25 straight \pi over denominator 2 square root of 2 end fraction plus fraction numerator 25 straight \pi over denominator 2 square root of 2 end fraction plus fraction numerator 25 straight \pi over denominator 2 square root of 2 end fraction plus fraction numerator 25 straight \pi over denominator 2 square root of 2 end fraction plus fraction numerator 25 straight \pi over denominator 2 square root of 2 end fraction plus fraction numerator 25 straight \pi over denominator 2 square root of 2 end fraction plus
plus fraction numerator 25 straight \pi over denominator 2 square root of 2 end fraction plus fraction numerator 25 straight \pi over denominator 2 square root of 2 end fraction right parenthesis equals 1 half times 8 times fraction numerator 25 straight \pi over denominator 2 square root of 2 end fraction equals fraction numerator 50 straight \pi over denominator square root of 2 end fraction equals fraction numerator 50 straight \pi over denominator square root of 2 end fraction fraction numerator square root of 2 over denominator square root of 2 end fraction equals fraction numerator 50 square root of 2 straight \pi over denominator 2 end fraction equals
equals 25 square root of 2 straight pi


     

    Ewentualnie sposób alternatywny.

    Ponieważ (patrz rysunek) figura jest symetryczna względem obu osi współrzędnych, to możemy obliczyć pole jednej ćwierci (należącej do pierwszej ćwiartki) i wynik pomnożyć razy 4. Wtedy nie będzie potrzeby liczyć całki niewłaściwe. Czyli:

    P equals 1 half integral subscript 0 superscript 2 \pi end superscript fraction numerator 100 over denominator 3 plus cos 4 phi end fraction d phi equals 4 times 1 half integral subscript 0 superscript \pi over 2 end superscript fraction numerator 100 over denominator 3 plus cos 4 phi end fraction d phi equals horizontal ellipsis

    • Ta odpowiedź została zmodyfikowana 1 rok, 7 miesiące temu przez Joanna Grochowska.
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