To set up the integral for the mass of the solid bounded by the surfaces $z = \sqrt{16 - x^2 - y^2}$, the $xy$-plane ($z = 0$), and the $xz$-plane ($y = 0$), with density function $f(x, y, z) = x^2 y^2 z^2$, we’ll follow these steps:

1. Analyze the Boundaries

• The surface $z = \sqrt{16 - x^2 - y^2}$ represents the upper boundary of the solid, meaning $z$ ranges from $0$ to $\sqrt{16 - x^2 - y^2}$.
• The region in the $xy$-plane that this solid projects onto is the circular region defined by $x^2 + y^2 \leq 16$.
• Since we are bounded by the $xz$-plane ($y = 0$), the variable $y$ will range from $0$ to $\sqrt{16 - x^2}$.

2. Set Up the Triple Integral

The mass $M$ of the solid is given by the integral: $M = \iiint\limits_{\text{solid}} f(x, y, z) \, dV$ where $f(x, y, z) = x^2 y^2 z^2$.

In Cartesian Coordinates:

We can set up the integral in Cartesian coordinates: $M = \int_{x=-4}^{4} \int_{y=0}^{\sqrt{16 - x^2}} \int_{z=0}^{\sqrt{16 - x^2 - y^2}} x^2 y^2 z^2 \, dz \, dy \, dx$ where:

• $z$ ranges from $0$ to $\sqrt{16 - x^2 - y^2}$,
• $y$ ranges from $0$ to $\sqrt{16 - x^2}$,
• $x$ ranges from $-4$ to $4$.

3. Alternative in Cylindrical Coordinates

Alternatively, we can use cylindrical coordinates:

• Set $x = r \cos \theta$, $y = r \sin \theta$, and $z = z$.
• The density function becomes $f(x, y, z) = (r \cos \theta)^2 (r \sin \theta)^2 z^2 = r^4 \cos^2 \theta \sin^2 \theta z^2$.
• The region for $r$ is $0 \leq r \leq 4$, $\theta$ is $0 \leq \theta \leq \frac{\pi}{2}$, and $z$ is $0 \leq z \leq \sqrt{16 - r^2}$.

Thus, the integral in cylindrical coordinates is: $M = \int_{\theta=0}^{\pi/2} \int_{r=0}^{4} \int_{z=0}^{\sqrt{16 - r^2}} r^4 \cos^2 \theta \sin^2 \theta \, z^2 \, r \, dz \, dr \, d\theta$ where we include an additional $r$ from the Jacobian determinant.

This setup provides the integral required to find the mass of the solid.

Would you like further details or have any questions on this setup?

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Related Questions

1. How do we evaluate the integral for mass step-by-step?
2. How can we set up similar integrals for solids bounded by other parabolic surfaces?
3. What are the benefits of using cylindrical over Cartesian coordinates in such integrals?
4. How do boundary planes affect the limits of integration in multivariable calculus?
5. What changes if the density function is dependent only on one variable, say $f(x, y, z) = x^2$?

Tip

In triple integrals, always check the geometry and symmetry of the solid, as these can simplify both the setup and evaluation.