integral from-2 to 2 of sqrt(8-x^2)

Lösung

\int_{- 2}^{2} 8 - x^{2} d x

4 (\frac{π}{2} + 1)

Dezimale

10.28318 \dots

Schritte zur Lösung

\int_{- 2}^{2} 8 - x^{2} d x

Trigonometrische Substitution anwenden

= \int_{- \frac{π}{4}}^{\frac{π}{4}} 8 cos^{2} (u) d u

Entferne die Konstante:

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

= 8 \cdot \int_{- \frac{π}{4}}^{\frac{π}{4}} cos^{2} (u) d u

Umschreiben mit Hilfe von Trigonometrie-Identitäten

= 8 \cdot \int_{- \frac{π}{4}}^{\frac{π}{4}} \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

= 8 \cdot \frac{1}{2} \cdot \int_{- \frac{π}{4}}^{\frac{π}{4}} 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

= 8 \cdot \frac{1}{2} (\int_{- \frac{π}{4}}^{\frac{π}{4}} 1 d u + \int_{- \frac{π}{4}}^{\frac{π}{4}} cos (2 u) d u)

\int_{- \frac{π}{4}}^{\frac{π}{4}} 1 d u = \frac{π}{2}

\int_{- \frac{π}{4}}^{\frac{π}{4}} cos (2 u) d u = 1

= 8 \cdot \frac{1}{2} (\frac{π}{2} + 1)

Vereinfache

8 \cdot \frac{1}{2} (\frac{π}{2} + 1) : 4 (\frac{π}{2} + 1)

= 4 (\frac{π}{2} + 1)

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