<s>[INST] Evaluate the triple integral $\int_0^2\int_0^y\int_0^{\sqrt{4-y^2}}2xdxdzdy\\$. $$
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\int_0^2\int_0^y\int_0^{\sqrt{4-y^2}}2xdxdzdy\\
\int_0^2\int_0^y4-y^2dzdy\\
\int_0^2(4-y^2)ydy\\
-\frac{1}{2}\left( \left.4y-\frac{1}{3}y^3\right|_0^2\right )\\
=-\frac{8}{3}
$$
I have worked over this problem several times and I cannot find the step where I went wrong.
 [/INST] The step where you made mistake is in evaluating the integral $\displaystyle \int_0^2 (4-y^2)ydy$
$$\int_0^2 (4-y^2)ydy = \int_0^2 \left( 4y - y^3\right)dy = \left(2y^2 - y^4/4 \right)_{y=0}^{y=2} = 8 - 2^4/4 = 8 - 4 =4$$
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