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I have to find the region $E$ where

$ \iiint_E (1-x^2 - 2y^2 - 3z^2) dV $

has maximum value, but I'm not sure how to start.

I was thinking of getting the derivative of the integral and then finding the extrema with the usual $f_x = 0, f_y = 0$ equation system, but even if I get the second derivative of it, I'm stuck with a single integral and I'm not sure if that's right.

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    let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/3435/discussion-between-marvis-and-user1002327)2012-05-14

1 Answers 1

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Maximizing the integral means integrating over positive integrands. This means the region that maximizes the integral is

$ E = \left\{ (x,y,z) : 1 -x^2 - 2y^2 -3y^2 \geq 0\right\} \\ = \left\{ (x,y,z) : x^2 + 2y^2 + 3y^2 \leq 1 \right\} $