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:
- For z: from 0 to 1 - y²
- For y: from x² to 1
- 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.