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Let W be the region bounded by z = 1 - y², y = x² and the plane z = 0. Calculate the volume of W in the order dz dy dx.

User Boleto
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1 Answer

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Final answer:

The volume of region W is calculated using a triple integral with limits determined by the surfaces z = 1 - y² and y = x², and the plane z = 0. Integrating from the inside out (dz, dy, dx), we find the volume within the given bounds.

Step-by-step explanation:

To calculate the volume of the region W described by the student, bounded by the surfaces z = 1 - y², y = x², and the plane z = 0, we need to set up a triple integral in the order dz dy dx. We evaluate the integral within the bounds established by the given equations:

  • The surface z = 1 - y² sets the upper limit for z.
  • The plane z = 0 sets the lower limit for z.
  • The parabola y = x² sets the limits for y, considering that y must be positive since it is a square of x.
  • Since z is bounded above by 1 - y², y ranges from 0 to 1, and thus x ranges from -1 to 1.

Therefore, our limits of integration are:

  1. For z: from 0 to 1 - y²
  2. For y: from to 1
  3. For x: from -1 to 1

Now, the triple integral for calculating the volume V is:


∫-1∫1∫x²∫1∫0∫1-y² dz dy dx

Evaluating the inner integral with respect to z first, then y, and finally x, gives us the final volume of W.

User Tyler Miller
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