integral from 0 to 6 of 2pi(y+2)(3-y/2)

Lösung

\int_{0}^{6} 2 π (y + 2) (3 - \frac{y}{2}) d y

72 π

Dezimale

226.19467 \dots

Schritte zur Lösung

\int_{0}^{6} 2 π (y + 2) (3 - \frac{y}{2}) d y

Entferne die Konstante:

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

= 2 π \cdot \int_{0}^{6} (y + 2) (3 - \frac{y}{2}) d y

Vereinfache

(y + 2) (3 - \frac{y}{2}) : - \frac{y ^{2}}{2} + 2 y + 6

= 2 π \cdot \int_{0}^{6} - \frac{y ^{2}}{2} + 2 y + 6 d y

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 π (- \int_{0}^{6} \frac{y ^{2}}{2} d y + \int_{0}^{6} 2 y d y + \int_{0}^{6} 6 d y)

\int_{0}^{6} \frac{y ^{2}}{2} d y = 36

\int_{0}^{6} 2 y d y = 36

\int_{0}^{6} 6 d y = 36

= 2 π (- 36 + 36 + 36)

Vereinfache

= 72 π