integral from 0 to 2 of ysqrt(2y-y^2)

Lösung

\int_{0}^{2} y 2 y - y^{2} d y

\frac{π}{2}

Dezimale

1.57079 \dots

Schritte zur Lösung

\int_{0}^{2} y 2 y - y^{2} d y

Vervollständige das Quadrat

2 y - y^{2} : - (y - 1)^{2} + 1

= \int_{0}^{2} y - (y - 1)^{2} + 1 d y

Wende U-Substitution an

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

Multipliziere aus

(u + 1) - u^{2} + 1 : u - u^{2} + 1 + - u^{2} + 1

= \int_{- 1}^{1} u - u^{2} + 1 + - u^{2} + 1 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

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

\int_{- 1}^{1} u - u^{2} + 1 d u = 0

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

= 0 + \frac{π}{2}

Vereinfache

= \frac{π}{2}

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