Evaluate the integral by changing the order of integration in an appropriate way.
Integral from 0 to 1 Integral from RootIndex 5 StartRoot z EndRoot to 1 Integral from 0 to ln 4 StartFraction pi e Superscript 2 x Baseline sine left parenthesis pi y squared right parenthesis Over y Superscript 4 EndFraction dx font size decreased by 6 dy font size decreased by 6 dz

To evaluate the given triple integral, we will first change the order of integration. The integral is:

\[
\int_0^1 \int_{\sqrt{z}}^1 \int_0^{\ln(4)} \frac{\pi e^{2x} \sin(\pi y^2)}{y^4} \, dx \, dy \, dz
\]

### Step 1: Identify the region of integration
The bounds of integration are:

- \( z \in [0, 1] \)
- For a given \( z \), \( y \in [\sqrt{z}, 1] \)
- For a given \( y \), \( x \in [0, \ln(4)] \)

The integral order is currently \( dz \, dy \, dx \). We want to change this order to facilitate easier computation.

### Step 2: Express the bounds differently

Currently, the region is defined by:

- \( z = y^2 \) (from \( z = \sqrt{z} \leq y \leq 1 \))
- \( y \in [0, 1] \)
- \( x \in [0, \ln(4)] \)

To reverse the integration order, we now express \( z \) in terms of \( y \), which is:

- \( z \in [0, y^2] \)

### Step 3: Change the order of integration

When changing the order of integration, the new bounds become:

- \( y \in [0, 1] \)
- \( z \in [0, y^2] \)
- \( x \in [0, \ln(4)] \)

So the integral becomes:

\[
\int_0^1 \int_0^{y^2} \int_0^{\ln(4)} \frac{\pi e^{2x} \sin(\pi y^2)}{y^4} \, dx \, dz \, dy
\]

### Step 4: Evaluate the inner integrals

1. **Integrating with respect to \( x \):**

   The integrand with respect to \( x \) is \( \pi e^{2x} \sin(\pi y^2) / y^4 \). The \( x \)-integral is:

   \[
   \int_0^{\ln(4)} e^{2x} \, dx = \left[ \frac{e^{2x}}{2} \right]_0^{\ln(4)} = \frac{e^{2\ln(4)} - e^0}{2} = \frac{16 - 1}{2} = \frac{15}{2}
   \]

   Thus, after integrating with respect to \( x \), the integral becomes:

   \[
   \frac{15\pi}{2} \int_0^1 \int_0^{y^2} \frac{\sin(\pi y^2)}{y^4} \, dz \, dy
   \]

2. **Integrating with respect to \( z \):**

   Since \( z \) does not appear in the integrand, the \( z \)-integral is straightforward:

   \[
   \int_0^{y^2} dz = y^2
   \]

   After this, the integral becomes:

   \[
   \frac{15\pi}{2} \int_0^1 \frac{y^2 \sin(\pi y^2)}{y^4} \, dy = \frac{15\pi}{2} \int_0^1 \frac{\sin(\pi y^2)}{y^2} \, dy
   \]

### Step 5: Substitution for the remaining integral

Let \( u = y^2 \), so \( du = 2y \, dy \), and the limits of integration change accordingly:

- When \( y = 0 \), \( u = 0 \)
- When \( y = 1 \), \( u = 1 \)

Thus, the integral becomes:

\[
\frac{15\pi}{2} \int_0^1 \frac{\sin(\pi u)}{u} \, \frac{du}{2} = \frac{15\pi}{4} \int_0^1 \frac{\sin(\pi u)}{u} \, du
\]

### Step 6: Standard result

The integral \( \int_0^1 \frac{\sin(\pi u)}{u} \, du \) is a standard integral and equals \( \frac{\pi}{2} \).

Thus, the final result is:

\[
\frac{15\pi}{4} \times \frac{\pi}{2} = \frac{15\pi^2}{8}
\]

### Final Answer:
The value of the integral is:

\[
\boxed{\frac{15\pi^2}{8}}
\]

Let me know if you'd like further details on any of these steps!

---

Here are five related questions for deeper exploration:

1. How do you determine when changing the order of integration simplifies a problem?
2. What other techniques, besides changing the order, can help with solving complex integrals?
3. Can you explain the geometrical meaning of changing the order of integration in multiple integrals?
4. What are the conditions for interchanging the limits of integration in double or triple integrals?
5. How do improper integrals relate to this kind of problem?

**Tip:** Always check if simplifying the bounds of integration through coordinate transformation (polar, cylindrical, spherical, etc.) makes the problem easier to solve!

Evaluate the integral by changing the order of integration in an appropriate way. Integral from 0 to 1 Integral from RootIndex 5 StartRoot z EndRoot to 1 Integral from 0 to ln 4 StartFraction pi e Superscript 2 x Baseline sine left parenthesis pi y squared right parenthesis Over y Superscript 4 EndFraction dx font size decreased by 6 dy font size decreased by 6 dz

This detailed solution walks through evaluating a triple integral by changing the order of integration. The problem involves concepts of definite integrals, substitution, and Fubini's theorem, culminating in a final result of \( \frac{15\pi^2}{8} \). Ideal for students studying multivariable calculus.

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