Find the volume of the given solid region in the first octant bounded by the plane 12 x plus 20 y plus 15 z equals 60 and the coordinate​ planes, using triple integrals.
left parenthesis 5 comma 0 comma 0 right parenthesis
left parenthesis 0 comma 3 comma 0 right parenthesis
left parenthesis 0 comma 0 comma 4 right parenthesis
Question content area bottom
Part 1
Complete the triple integral below used to find the volume of the given solid region. Note the order of integration dz dy dx.
VequalsIntegral from 0 to nothing Integral from 0 to nothing Integral from 0 to nothing dz dy dx
  
  
  


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.

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 triple integrals. (5, 0, 0), (0, 3, 0), (0, 0, 4). Complete the triple integral dz dy dx.

Learn how to find the volume of a solid region bounded by the plane 12x + 20y + 15z = 60 in the first octant using triple integrals. This solution involves setting up the correct bounds for dz, dy, and dx based on the plane intersections. Suitable for college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation