integral from 0 to 3 of 9*(9-y^2)^{1/2}

Lösung

\int_{0}^{3} 9 \cdot (9 - y^{2})^{\frac{1}{2}} d y

\frac{81 π}{4}

Dezimale

63.61725 \dots

Schritte zur Lösung

\int_{0}^{3} 9 (9 - y^{2})^{\frac{1}{2}} d y

Entferne die Konstante:

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

= 9 \cdot \int_{0}^{3} (9 - y^{2})^{\frac{1}{2}} d y

Trigonometrische Substitution anwenden

= 9 \cdot \int_{0}^{\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 9 \cdot \int_{0}^{\frac{π}{2}} cos^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 9 \cdot 9 \cdot \int_{0}^{\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 9 \cdot \frac{1}{2} \cdot \int_{0}^{\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 9 \cdot \frac{1}{2} (\int_{0}^{\frac{π}{2}} 1 d u + \int_{0}^{\frac{π}{2}} cos (2 u) d u)

\int_{0}^{\frac{π}{2}} 1 d u = \frac{π}{2}

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

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

Vereinfache

9 \cdot 9 \cdot \frac{1}{2} (\frac{π}{2} + 0) : \frac{81 π}{4}

= \frac{81 π}{4}

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