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To express the integral \(\iiint_E f(x, y, z) \, dV\) as an iterated integral in six different ways, given that \(E\) is bounded by \(z = 0\), \(z = 2y\), and \(x^2 = 25 - y\), let's analyze each bounding function to set up the integrals.

### Region Analysis

1. **Boundaries for \(z\):**
   - \(z\) ranges from \(0\) (the plane \(z = 0\)) to \(2y\) (the plane \(z = 2y\)).

2. **Boundaries for \(y\):**
   - The condition \(x^2 = 25 - y\) defines a parabola. Solving for \(y\):
     \[
     y = 25 - x^2
     \]
   - Therefore, \(y\) ranges from \(0\) to \(25\) when \(x = 0\), and for a given \(x\), it ranges from \(0\) to \(25 - x^2\).

3. **Boundaries for \(x\):**
   - Since \(y\) must be non-negative, the values of \(x\) satisfy \(-5 \leq x \leq 5\).

### Writing Six Different Iterated Integrals

1. **Integral with \(dz \, dy \, dx\):**
   \[
   \int_{-5}^{5} \int_{0}^{25 - x^2} \int_{0}^{2y} f(x, y, z) \, dz \, dy \, dx
   \]

2. **Integral with \(dz \, dx \, dy\):**
   \[
   \int_{0}^{25} \int_{-\sqrt{25 - y}}^{\sqrt{25 - y}} \int_{0}^{2y} f(x, y, z) \, dz \, dx \, dy
   \]

3. **Integral with \(dx \, dy \, dz\):**
   \[
   \int_{0}^{50} \int_{0}^{y / 2} \int_{-\sqrt{25 - y}}^{\sqrt{25 - y}} f(x, y, z) \, dx \, dy \, dz
   \]

4. **Integral with \(dx \, dz \, dy\):**
   \[
   \int_{0}^{50} \int_{0}^{25 - x^2} \int_{0}^{2y} f(x, y, z) \, dx \, dz \, dy
   \]

5. **Integral with \(dy \, dz \, dx\):**
   \[
   \int_{0}^{50} \int_{-5}^{5} \int_{0}^{25 - x^2} f(x, y, z) \, dy \, dz \, dx
   \]

6. **Integral with \(dy \, dx \, dz\):**
   \[
   \int_{-5}^{5} \int_{0}^{2y} \int_{0}^{25 - y} f(x, y, z) \, dy \, dx \, dz

Express the integral \(\iiint_E f(x, y, z) \, dV\) as an iterated integral in six different ways, where \(E\) is the solid bounded by \(z = 0\), \(z = 2y\), and \(x^2 = 25 - y\).

Learn how to express a triple integral \(\iiint_E f(x, y, z) \, dV\) in six different iterated forms, where \(E\) is bounded by \(z = 0\), \(z = 2y\), and \(x^2 = 25 - y\). This problem covers multivariable calculus concepts, focusing on triple integrals and region boundaries.

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