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To find the volume of the solid \( R \) bounded by the given surfaces, we need to set up the iterated integral with appropriate limits of integration for \( x \), \( y \), and \( z \).

### Problem Summary
1. **Surfaces defining \( z \):**
   - **Plane \( z = 3x \)**
   - **Surface \( z = x^2 \)**

2. **Planes defining \( y \):**
   - **Plane \( y = 0 \)**
   - **Plane \( y = 4 \)**

The solid \( R \) is bounded by these surfaces and planes.

### Steps to Determine Limits of Integration
1. **Identify \( x \) limits:**
   - To find the limits for \( x \), observe where \( z = 3x \) and \( z = x^2 \) intersect. Set \( 3x = x^2 \).
   - Solving \( x^2 - 3x = 0 \), we factor as \( x(x - 3) = 0 \), giving \( x = 0 \) and \( x = 3 \).
   - Therefore, the range for \( x \) is \( 0 \leq x \leq 3 \).

2. **Identify \( y \) limits:**
   - The limits for \( y \) are given directly by the planes: \( 0 \leq y \leq 4 \).

3. **Identify \( z \) limits:**
   - For a fixed \( x \), \( z \) ranges from the lower surface \( z = x^2 \) to the upper surface \( z = 3x \).
   - So, the limits for \( z \) are \( x^2 \leq z \leq 3x \).

### Setting up the Integral
Now that we have the limits for each variable, we can write the iterated integral for the volume of \( R \):

\[
\iiint_R f(x, y, z) \, dV = \int_{x=0}^{3} \int_{y=0}^{4} \int_{z=x^2}^{3x} 1 \, dz \, dy \, dx.
\]

Here, \( f(x, y, z) = 1 \) since we are computing the volume.

### Final Answer
\[
\int_{x=0}^{3} \int_{y=0}^{4} \int_{z=x^2}^{3x} 1 \, dz \, dy \, dx.
\]

Would you like a detailed explanation of any part of this solution?

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### Related Questions:
1. How do we determine the intersection points between two surfaces in 3D space?
2. Why is the function \( f(x, y, z) \) set to \( 1 \) when calculating volume?
3. Can we change the order of integration, and if so, how does it affect the limits?
4. What if \( R \) was bounded by different surfaces for \( z \)? How would that change the integral setup?
5. How does integrating over a volume differ from integrating over an area?

### Tip:
When setting up triple integrals for volumes, start by finding the bounds for each variable, often by examining intersections and constraints.

Suppose R is the solid bounded by the plane z = 3x, the surface z = x^2, and the planes y = 0 and y = 4. Write an iterated integral in the form below to find the volume of the solid R with limits of integration.

Learn how to set up an iterated integral to find the volume of a solid bounded by the surfaces z = 3x and z = x^2, with planes y = 0 and y = 4. This example explores concepts in multivariable calculus, including volume calculation through triple integrals.

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