integral from 0 to 1/2 of sqrt(81-324x^2)

Lösung

\int_{0}^{\frac{1}{2}} 81 - 324 x^{2} d x

\frac{9 π}{8}

Dezimale

3.53429 \dots

Schritte zur Lösung

\int_{0}^{\frac{1}{2}} 81 - 324 x^{2} d x

Trigonometrische Substitution anwenden

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

Entferne die Konstante:

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \frac{9}{2} \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

= \frac{9}{2} \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

= \frac{9}{2} \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

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

Vereinfache

\frac{9}{2} \cdot \frac{1}{2} (\frac{π}{2} + 0) : \frac{9 π}{8}

= \frac{9 π}{8}

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