Evaluate the triple integral.
$$
\int_{1}^{2} \int_{1}^{z} \int_{-y}^{y} 2 x-z d x d y d z=
$$

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Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ We start by integrating the function $2x - z$ with respect to $x$ from $-y$ to $y$. ‖ ‖ step 2 ⋮ The result of the integration is $-2yz$, as the integral of $2x$ from $-y$ to $y$ is $0$ and the integral of $-z$ is $-2yz$. ‖ ‖ step 3 ⋮ Next, we integrate the result $-2yz$ with respect to $y$ from $1$ to $z$. ‖ ‖ step 4 ⋮ The result of this integration is $x(z - 1)(x - z)$. ‖ ‖ step 5 ⋮ Finally, we integrate the result $x(z - 1)(x - z)$ with respect to $z$ from $1$ to $2$. ‖ ‖ step 6 ⋮ The final result of the triple integral is $(x^4 z)/2 - x^4/2 - (x^3 z^2)/2 - (x^3 z)/3 + (5 x^3)/6 + (5 x^2 z^2)/6 - (5 x^2 z)/6$. ‖ ∻[question number] Answer∻ ⚹ $(x^4 z)/2 - x^4/2 - (x^3 z^2)/2 - (x^3 z)/3 + (5 x^3)/6 + (5 x^2 z^2)/6 - (5 x^2 z)/6$ ⚹ ∻Key Concept∻ ⚹Triple Integration⚹ ∻Explanation∻ ⚹The triple integral of a function over a given region is evaluated by performing three successive integrations, each with respect to one variable, while considering the limits of integration for each variable.⚹◊What is the result of evaluating the triple integral $\int_{1}^{2} \int_{1}^{z} \int_{-y}^{y} (2x - z) dx dy dz$?⍭ Generate me a similar question◊

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