Question-215604

Question Number 215604 by universe last updated on 11/Jan/25

Commented by MathematicalUser2357 last updated on 12/Jan/25

c a n y o u p l e a s e t r a n s l a t e

Answered by mr W last updated on 12/Jan/25

= \frac{1}{π} \int \int \int (x^{2} + 4 y^{2} + z^{2} - 2 x y - 4 y z + 2 z x + 4 x^{2} + y^{2} + z^{2} - 4 x y + 2 y z - 4 z x + x^{2} + y^{2} + 4 z^{2} - 2 x y - 4 y z + 4 z x) d x d y d z

= \frac{1}{π} \int \int \int (6 x^{2} + 6 y^{2} + 6 z^{2} - 8 x y - 6 y z + 2 z x) d x d y d z

= \frac{6}{π} \int \int \int (x^{2} + y^{2} + z^{2}) d x d y d z

= \frac{6}{π} \int_{0}^{1} r^{2} 4 π r^{2} d r

= 24 \times {[\frac{r^{5}}{5}]}_{0}^{1}

= 24 \times \frac{1^{5}}{5}

= 4.80 ✓

Commented by mr W last updated on 12/Jan/25

d u e t o s y m m e t r y

\underset{x^{2} + y^{2} + z^{2} ⩽ R^{2}}{\int \int \int} x y d x d y d z = 0

\underset{x^{2} + y^{2} + z^{2} ⩽ R^{2}}{\int \int \int} y z d x d y d z = 0

\underset{x^{2} + y^{2} + z^{2} ⩽ R^{2}}{\int \int \int} z x d x d y d z = 0

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