To find the mass of the solid bounded by the given surfaces and with the given density function, we need to set up and evaluate a triple integral.

Step 1: Determine the Volume Boundaries

The solid is bounded by the surfaces:

1. $y = 6x^2$
2. $y = 8$
3. $z = 0$
4. $z = 7$

The region is in the first octant where $x \geq 0$, $y \geq 0$, and $z \geq 0$.

Step 2: Density Function

The density function at a point $P(x, y, z)$ is given by $\rho = 5x$, because the distance to the yz-plane is $x$.

Step 3: Setting Up the Triple Integral

The mass $M$ of the solid is given by: $M = \iiint_V \rho \, dV$

Since the density $\rho = 5x$, we have: $M = \iiint_V 5x \, dV$

Step 4: Limits of Integration

• $x$ ranges from $0$ to $\sqrt{\frac{y}{6}}$ because $y = 6x^2$.
• $y$ ranges from $0$ to $8$.
• $z$ ranges from $0$ to $7$.

Therefore, the triple integral becomes: $M = \int_{0}^{8} \int_{0}^{\sqrt{\frac{y}{6}}} \int_{0}^{7} 5x \, dz \, dx \, dy$

Step 5: Evaluating the Integral

First, integrate with respect to $z$: $\int_{0}^{7} 5x \, dz = 5x \left[ z \right]_{0}^{7} = 5x(7 - 0) = 35x$

Now, substitute this result into the remaining integral: $M = \int_{0}^{8} \int_{0}^{\sqrt{\frac{y}{6}}} 35x \, dx \, dy$

Next, integrate with respect to $x$: $\int_{0}^{\sqrt{\frac{y}{6}}} 35x \, dx = 35 \left[ \frac{x^2}{2} \right]_{0}^{\sqrt{\frac{y}{6}}} = 35 \left( \frac{(\sqrt{\frac{y}{6}})^2}{2} - 0 \right) = 35 \left( \frac{y/6}{2} \right) = \frac{35y}{12}$

Finally, integrate with respect to $y$: $M = \int_{0}^{8} \frac{35y}{12} \, dy = \frac{35}{12} \int_{0}^{8} y \, dy$ $\frac{35}{12} \left[ \frac{y^2}{2} \right]_{0}^{8} = \frac{35}{12} \left( \frac{8^2}{2} - 0 \right) = \frac{35}{12} \left( \frac{64}{2} \right) = \frac{35}{12} \times 32 = \frac{1120}{12} = 93.33\overline{3}$

Final Answer

The mass of the solid is: $M = 93.\overline{3} \, \text{units of mass}$

Do you want any details or have any questions?

Here are some related questions:

1. How do you set up triple integrals for finding volumes?
2. What are the common methods for integrating functions of several variables?
3. How do you interpret the density function in physical problems?
4. Can you describe the method of slicing to find volumes of solids?
5. What are the applications of triple integrals in real-world problems?
6. How do you determine the limits of integration in cylindrical coordinates?
7. What are the differences between Cartesian, cylindrical, and spherical coordinate systems?
8. How do you handle integrals with variable limits?

Tip: When solving problems involving density functions and volumes, visualize the region and carefully determine the bounds of integration to set up the correct integral.