integral from y^2 to 8-2x^2-y^2 of 10xz

解

\int_{y^{2}}^{8 - 2 x^{2} - y^{2}} 10 x z d z

20 x (x^{4} + x^{2} y^{2} - 8 x^{2} + 16 - 4 y^{2})

解答ステップ

\int_{y^{2}}^{8 - 2 x^{2} - y^{2}} 10 x z d z

定数を除く:

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

= 10 x \cdot \int_{y^{2}}^{8 - y^{2} - 2 x^{2}} z d z

べき乗則を適用する

= 10 x [\frac{z ^{2}}{2}]_{y^{2}}^{8 - y^{2} - 2 x^{2}}

限度を計算する:

\frac{( 8 - y ^{2} - 2 x ^{2} ) ^{2}}{2} - \frac{y ^{4}}{2}

= 10 x (\frac{( 8 - y ^{2} - 2 x ^{2} ) ^{2}}{2} - \frac{y ^{4}}{2})

簡素化

= 20 x (x^{4} + x^{2} y^{2} - 8 x^{2} + 16 - 4 y^{2})