x-y-z-x-2-y-2-1-x-0-0-lt-z-lt-y-1-calculer-I-xyzdxdydz-please-i-need-help-

Question Number 195995 by pticantor last updated on 15/Aug/23

Δ = {(\bar{x} y z), x^{2} + y^{2} ⩽ 1, x ⩾ 0, 0 < z < y + 1}

c a l c u l e r I = \int \int \int_{Δ} x y z d x d y d z

p l e a s e i n e e d h e l p

Answered by aleks041103 last updated on 27/Aug/23

I = \int_{0}^{1} x \int_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} y \int_{0}^{y + 1} z d z d y d x

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

\int_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} y \int_{0}^{y + 1} z d z d y = \frac{1}{2} \int_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} y {(y + 1)}^{2} d y =

= \frac{1}{2} \int_{- \sqrt{1 - x^{2}}}^{\sqrt{1 - x^{2}}} (y^{3} + 2 y^{2} + y) d y =

= \frac{1}{2} {[\frac{y^{4}}{4} + \frac{2 y^{3}}{3} + \frac{y^{2}}{2}]}_{- s}^{s} =

= \frac{2}{3} s^{3} = \frac{2}{3} {(1 - x^{2})}^{3 / 2}

\Rightarrow I = \frac{1}{3} \int_{0}^{1} {(1 - x^{2})}^{3 / 2} 2 x d x =

= \frac{1}{3} \int_{0}^{1} {(1 - t)}^{3 / 2} d t =

= \frac{1}{3} \int_{0}^{1} r^{3 / 2} d r = \frac{1}{3} \frac{r^{5 / 2}}{5 / 2} = \frac{2}{15}

\Rightarrow I = \frac{2}{15}