To find the volume of the given solid region in the first octant bounded by the plane $12x + 20y + 15z = 60$ and the coordinate planes using a triple integral, we need to set up the integral properly.

Step 1: Rewrite the plane equation for bounds

The equation of the plane is: $12x + 20y + 15z = 60$ Let's solve for $z$ in terms of $x$ and $y$: $z = \frac{60 - 12x - 20y}{15} = 4 - \frac{4x}{5} - \frac{4y}{3}$

Step 2: Determine the limits of integration

Since we are considering the first octant, the limits for $x$, $y$, and $z$ are from 0 to their respective bounds:

• For $x$, the intersection with the $x$-axis gives $x = 5$ when $y = 0$ and $z = 0$.
• For $y$, the intersection with the $y$-axis gives $y = 3$ when $x = 0$ and $z = 0$.
• For $z$, it ranges from 0 to $4 - \frac{4x}{5} - \frac{4y}{3}$.

Step 3: Set up the triple integral

The volume $V$ can be found using the triple integral $\int \int \int \, dz \, dy \, dx$.

$V = \int_{0}^{5} \int_{0}^{3 - \frac{3x}{5}} \int_{0}^{4 - \frac{4x}{5} - \frac{4y}{3}} dz \, dy \, dx$

Step 4: Complete the integral limits

• The outermost integral with respect to $x$ is from $0$ to $5$.
• The middle integral with respect to $y$ is from $0$ to $3 - \frac{3x}{5}$.
• The innermost integral with respect to $z$ is from $0$ to $4 - \frac{4x}{5} - \frac{4y}{3}$.

Thus, the complete triple integral is: $V = \int_{0}^{5} \int_{0}^{3 - \frac{3x}{5}} \int_{0}^{4 - \frac{4x}{5} - \frac{4y}{3}} dz \, dy \, dx$

Would you like a step-by-step solution to evaluate this integral?

Related Questions:

1. What is the geometric interpretation of a triple integral in this context?
2. How would you evaluate the volume if the plane equation was different?
3. How does the order of integration (e.g., $dz \, dx \, dy$) affect the computation?
4. What are the limits of integration for a different solid region bounded by another plane?
5. Can we use cylindrical or spherical coordinates for this volume calculation?

Tip:

When setting up triple integrals for volumes, always sketch the region and understand the plane intersections to set accurate bounds.