integral from 0 to 10 of xsqrt(100-x^2)

Lösung

\int_{0}^{10} x 100 - x^{2} d x

\frac{1000}{3}

Dezimale

333.33333 \dots

Schritte zur Lösung

\int_{0}^{10} x 100 - x^{2} d x

Wende U-Substitution an

= \int_{100}^{0} - \frac{u}{2} d u

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

= - \int_{0}^{100} - \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}^{100} u d u)

Wende die Potenzregel an

= - (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{100})

Vereinfache

= \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{100}

Berechne die Grenzen:

\frac{2000}{3}

= \frac{1}{2} \cdot \frac{2000}{3}

Vereinfache

= \frac{1000}{3}

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