Question

Use a computer algebra system to find the volume of the solid bounded by the graphs of the equations. $$ z=9-x^{2}-y^{2}, z=0 $$

Short answer

The volume of the solid V can be obtained by computing the double integral \(\int_0^{2 \pi} \int_0^{3} (9-r^2) \cdot r \, dr \, d\theta\). Compute this double integral using a computer algebra system.

Step-by-step solution

Identify the bounds

Identify the bounds for the integration. We are looking for where the paraboloid intersects with the xy-plane, i.e., when \(z = 0\). Solve the equation \(0 = 9 - x^{2} - y^{2}\) by rearranging it, we get \(x^{2} + y^{2} = 9\), this is a circle with radius 3 in the xy-plane. Thus, the region of integration is a circle with radius 3.

Set up the integral

To find the volume, integrate the function over the region by setting up the correct integral. Since the region of integration is a circle, it's easier to use polar coordinates, where \(x = r \cos(\theta)\), \(y = r \sin(\theta)\). The function becomes \(z = 9 - r^2\), where \(0 <= r <= 3\) and \(0 <= \theta <= 2\pi\). The volume of the solid V is given by the double integral \(\int_0^{2 \pi} \int_0^{3} (9-r^2) \cdot r \, dr \, d\theta\). Note the multiplication by r to account for the determinant of the Jacobian in polar coordinates.

Compute the integral

Now compute the double integral to calculate the volume. This is where we use a computer algebra system as requested in the exercise.