integral from 0 to 3 of (xsqrt(9-x^2))

Lösung

\int_{0}^{3} (x 9 - x^{2}) d x

9

Schritte zur Lösung

\int_{0}^{3} x 9 - x^{2} d x

Wende U-Substitution an

= \int_{9}^{0} - \frac{u}{2} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{0}^{9} - \frac{u}{2} d u

Entferne die Konstante:

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

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

Wende die Potenzregel an

= - (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{9})

Vereinfache

= \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{9}

Berechne die Grenzen:

18

= \frac{1}{2} \cdot 18

Vereinfache

= 9

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