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Schimbarea de variabilă la integrale definite

Schimbarea de variabilă pentru integralele definite este foarte asemănătoare cu cea pentru integralele nedefinite, dar se adaugă un pas suplimentar: adaptarea limitelor de integrare. Să vedem ce înseamnă asta calculând integral, start subscript, 1, end subscript, squared, start color #7854ab, 2, x, end color #7854ab, start color #e07d10, left parenthesis, end color #e07d10, start color #1fab54, x, squared, plus, 1, end color #1fab54, start color #e07d10, right parenthesis, cubed, end color #e07d10, start color #7854ab, d, x, end color #7854ab.
Observăm că start color #7854ab, 2, x, end color #7854ab este derivata lui start color #1fab54, x, squared, plus, 1, end color #1fab54, deci aplicăm schimbarea de variabilă. Dacă notăm start color #1fab54, u, equals, x, squared, plus, 1, end color #1fab54, atunci start color #7854ab, d, u, equals, 2, x, d, x, end color #7854ab. Acum înlocuim:
integral, start subscript, 1, end subscript, squared, start color #7854ab, 2, x, end color #7854ab, start color #e07d10, left parenthesis, end color #e07d10, start color #1fab54, x, squared, plus, 1, end color #1fab54, start color #e07d10, right parenthesis, cubed, end color #e07d10, start color #7854ab, d, x, end color #7854ab, equals, integral, start subscript, 1, end subscript, squared, start color #e07d10, left parenthesis, end color #e07d10, start color #1fab54, u, end color #1fab54, start color #e07d10, right parenthesis, cubed, end color #e07d10, start color #7854ab, d, u, end color #7854ab
Stai puțin! Limitele de integrare erau pentru x, nu pentru u. Să identificăm grafic. Ne interesa aria suprafeței cuprinse între curba start color #11accd, y, equals, 2, x, left parenthesis, x, squared, plus, 1, right parenthesis, cubed, end color #11accd de la x, equals, 1 până la x, equals, 2.
Function y = 2 x left parenthesis x squared + 1 right parenthesis cube is graphed. The x-axis goes from 0 to 3. The graph is a curve. The curve starts in quadrant 2, moves upward away from the x-axis to (2, 500). The region between the curve and the x-axis, between x = 1 and x = 2, is shaded.
Deoarece curba a devenit start color #aa87ff, y, equals, u, cubed, end color #aa87ff, nu mai putem păstra aceleași limite.
Functions y = 2 x left parenthesis x squared + 1 right parenthesis cube and y = u cubed are graphed together. The graph of y = u cubed starts in quadrant 2, moves upward away from the x-axis and ends at about (3, 27).
Sunt reprezentate grafic atât start color #11accd, y, equals, 2, x, left parenthesis, x, squared, plus, 1, right parenthesis, cubed, end color #11accd cât și start color #aa87ff, y, equals, u, cubed, end color #aa87ff. Se observă că ariile de sub curbe, cuprinse între x, equals, 1 și x, equals, 2 (sau u, equals, 1 și u, equals, 2) sunt foarte diferite.
Într-adevăr, limitele nu ar trebui să rămână la fel. Pentru a determina noile limite, trebuie să calculăm valorile lui start color #1fab54, u, end color #1fab54 care corespund lui start color #1fab54, x, squared, plus, 1, end color #1fab54 pentru x, equals, start color #ca337c, 1, end color #ca337c și x, equals, start color #ca337c, 2, end color #ca337c:
  • Limita inferioară: left parenthesis, start color #ca337c, 1, end color #ca337c, right parenthesis, squared, plus, 1, equals, start color #ca337c, 2, end color #ca337c
  • Limita superioară: left parenthesis, start color #ca337c, 2, end color #ca337c, right parenthesis, squared, plus, 1, equals, start color #ca337c, 5, end color #ca337c
Acum putem efectua corect schimbarea de variabilă:
integral, start subscript, start color #ca337c, 1, end color #ca337c, end subscript, start superscript, start color #ca337c, 2, end color #ca337c, end superscript, start color #7854ab, 2, x, end color #7854ab, start color #e07d10, left parenthesis, end color #e07d10, start color #1fab54, x, squared, plus, 1, end color #1fab54, start color #e07d10, right parenthesis, cubed, end color #e07d10, start color #7854ab, d, x, end color #7854ab, equals, integral, start subscript, start color #ca337c, 2, end color #ca337c, end subscript, start superscript, start color #ca337c, 5, end color #ca337c, end superscript, start color #e07d10, left parenthesis, end color #e07d10, start color #1fab54, u, end color #1fab54, start color #e07d10, right parenthesis, cubed, end color #e07d10, start color #7854ab, d, u, end color #7854ab
Functions y = 2 x left parenthesis x squared + 1 right parenthesis cube and y = u cubed are graphed together. The x-axis goes from negative 1 to 6. Each graph moves upward away from the x-axis. The first function ends at (2, 500). The region between the curve and the x-axis between x = 1 and x = 2 is shaded. The second function ends at about (6, 210). The region between the curve and the x-axis, between x = 1 and x = 5, is shaded. The 2 shaded regions look similar in size.
start color #aa87ff, y, equals, u, cubed, end color #aa87ff este reprezentat între u, equals, 2 și u, equals, 5. Acum putem vedea că suprafețele colorate au cam aceeași dimensiune (de fapt, sunt exact egale, dar e greu de spus doar din priviri).
De aici încolo, putem rezolva orice în raport de u:
25u3du=[u44]25=544244=152,25\begin{aligned} \displaystyle\int_{2}^5 u^3\,du&=\left[\dfrac{u^4}{4}\right]_{2}^5 \\\\ &=\dfrac{5^4}{4}-\dfrac{2^4}{4} \\\\ &=152{,}25 \end{aligned}
Amintește-ți: Când folosim schimbarea de variabilă pe integrale definite, trebuie întotdeauna să adaptăm limitele de integrare.
Problema 1
Elei i s-a cerut să determine integral, start subscript, 1, end subscript, start superscript, 5, end superscript, left parenthesis, 2, x, plus, 1, right parenthesis, left parenthesis, x, squared, plus, x, right parenthesis, cubed, d, x. Iată cum a lucrat ea:
Pasul 1: A notat u, equals, x, squared, plus, x
Pasul 2: d, u, equals, left parenthesis, 2, x, plus, 1, right parenthesis, d, x
Pasul 3:
integral, start subscript, 1, end subscript, start superscript, 5, end superscript, left parenthesis, 2, x, plus, 1, right parenthesis, left parenthesis, x, squared, plus, x, right parenthesis, cubed, d, x, equals, integral, start subscript, 1, end subscript, start superscript, 5, end superscript, u, cubed, d, u
Pasul 4:
15u3du=[u44]15=544144=156\begin{aligned} \displaystyle\int_1^5 u^3du&=\left[\dfrac{u^4}{4}\right]_1^5 \\\\ &=\dfrac{5^4}{4}-\dfrac{1^4}{4} \\\\ &=156 \end{aligned}
A lucrat corect? Dacă nu, care este greșeala ei?
Alege un răspuns:

Problema 2
integral, start subscript, 1, end subscript, squared, 15, x, squared, left parenthesis, x, cubed, minus, 7, right parenthesis, start superscript, 4, end superscript, d, x, equals, question mark
Alege un răspuns:

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