integral from 0 to 1 of rsqrt(2-r^2)

Lösung

\int_{0}^{1} r 2 - r^{2} d r

\frac{2 2 - 1}{3}

Dezimale

0.60947 \dots

Schritte zur Lösung

\int_{0}^{1} r 2 - r^{2} d r

Wende U-Substitution an

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

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

= - \int_{1}^{2} - \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_{1}^{2} u d u)

Wende die Potenzregel an

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

Vereinfache

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

Berechne die Grenzen:

\frac{4 2}{3} - \frac{2}{3}

= \frac{1}{2} (\frac{4 2}{3} - \frac{2}{3})

Vereinfache

= \frac{2 2 - 1}{3}

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