Suppose RR is the solid bounded by the plane z=4xz=4x, the surface z=x2z=x2, and the planes y=0y=0 and y=5y=5. Write an iterated integral in the form below to find the volume of the solid RR.

∭Rf(x,y,z)dV=∫BA∫DC∫FE∭Rf(x,y,z)dV=∫AB∫CD∫EF  dzdydxdzdydx
with limits of integration


To find the volume of the solid \( R \) bounded by the surfaces \( z = 4x \), \( z = x^2 \), and the planes \( y = 0 \), \( y = 5 \), we will set up an iterated integral. We aim to write this in the form:

\[
\iiint_R f(x, y, z) \, dV = \int_{A}^{B} \int_{C}^{D} \int_{E}^{F} \, dz \, dy \, dx
\]

### Step 1: Determine the Boundaries for \( z \)
The solid is bounded by the surfaces \( z = 4x \) and \( z = x^2 \) in terms of \( z \):
1. **Upper Bound**: \( z = 4x \)
2. **Lower Bound**: \( z = x^2 \)

So, for a fixed \( x \) and \( y \), \( z \) ranges from \( x^2 \) to \( 4x \).

Thus, we have:
\[
E = x^2 \quad \text{and} \quad F = 4x.
\]

### Step 2: Determine the Boundaries for \( y \)
The solid is also bounded by \( y = 0 \) and \( y = 5 \), so:
\[
C = 0 \quad \text{and} \quad D = 5.
\]

### Step 3: Determine the Boundaries for \( x \)
To find the limits for \( x \), we need to determine the points where \( z = 4x \) intersects \( z = x^2 \). Setting these equations equal to each other:

\[
4x = x^2
\]
\[
x^2 - 4x = 0
\]
\[
x(x - 4) = 0
\]

This gives \( x = 0 \) and \( x = 4 \).

So, the bounds for \( x \) are:
\[
A = 0 \quad \text{and} \quad B = 4.
\]

### Final Answer: Iterated Integral
The iterated integral to find the volume of \( R \) is:

\[
\iiint_R f(x, y, z) \, dV = \int_0^4 \int_0^5 \int_{x^2}^{4x} \, dz \, dy \, dx.
\]

This integral gives the volume when \( f(x, y, z) = 1 \).

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Would you like further details on how to solve the integral?

### Related Questions
1. How would the integral change if the plane \( y = 5 \) was replaced with \( y = 3 \)?
2. How do we interpret the geometric region bounded by \( z = 4x \) and \( z = x^2 \)?
3. What would the limits of integration be if we swapped the order of integration to \( dx \, dy \, dz \)?
4. How can we confirm that the region defined by these bounds represents a solid?
5. What role does the plane \( y = 0 \) play in defining the volume of the solid?

### Tip
For complex regions, sketching the boundaries can help visualize the order and range of integration.

Suppose RR is the solid bounded by the plane z=4x, the surface z=x^2, and the planes y=0 and y=5. Write an iterated integral in the form below to find the volume of the solid RR.

Learn how to set up and evaluate a triple integral to find the volume of a solid bounded by the planes z=4x, z=x^2, and y=0 to y=5. This problem involves using iterated integrals to calculate volume and applying the concept of boundaries in multivariable calculus. Suitable for students in Grades 11-12.

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