integral from-3 to 3 of 1sqrt(9-x^2)

Lösung

\int_{- 3}^{3} 1 9 - x^{2} d x

\frac{9 π}{2}

Dezimale

14.13716 \dots

Schritte zur Lösung

\int_{- 3}^{3} 1 \cdot 9 - x^{2} d x

9 - x^{2} = 9 (1 - \frac{x ^{2}}{9})

= \int_{- 3}^{3} 1 \cdot 9 (1 - \frac{x ^{2}}{9}) d x

Vereinfache

1 \cdot 9 (1 - \frac{x ^{2}}{9}) : 9 - x^{2}

= \int_{- 3}^{3} 9 - x^{2} d x

Trigonometrische Substitution anwenden

= \int_{- \frac{π}{2}}^{\frac{π}{2}} 9 cos^{2} (u) d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 9 \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} cos^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 9 \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{1 + cos ( 2 u )}{2} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 9 \cdot \frac{1}{2} \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + cos (2 u) d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 9 \cdot \frac{1}{2} (\int_{- \frac{π}{2}}^{\frac{π}{2}} 1 d u + \int_{- \frac{π}{2}}^{\frac{π}{2}} cos (2 u) d u)

\int_{- \frac{π}{2}}^{\frac{π}{2}} 1 d u = π

\int_{- \frac{π}{2}}^{\frac{π}{2}} cos (2 u) d u = 0

= 9 \cdot \frac{1}{2} (π + 0)

Vereinfache

9 \cdot \frac{1}{2} (π + 0) : \frac{9 π}{2}

= \frac{9 π}{2}

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