integral from 0 to a of b/a sqrt(a^2-x^2)

Lösung

\int_{0}^{a} \frac{b}{a} a^{2} - x^{2} d x

\frac{πba}{4}

Schritte zur Lösung

\int_{0}^{a} \frac{b}{a} a^{2} - x^{2} d x

Entferne die Konstante:

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

= \frac{b}{a} \cdot \int_{0}^{a} a^{2} - x^{2} d x

Trigonometrische Substitution anwenden

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

Entferne die Konstante:

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

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

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= \frac{b}{a} a^{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{b}{a} a^{2} \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{b}{a} a^{2} \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{b}{a} a^{2} \frac{1}{2} (\frac{π}{2} + 0)

Vereinfache

\frac{b}{a} a^{2} \frac{1}{2} (\frac{π}{2} + 0) : \frac{πba}{4}

= \frac{πba}{4}

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