To evaluate the triple integral given in the image:

$\iiint_E \frac{e^{-2x} \sin^2(z)}{\sqrt{y}} \, dV$

where $E = [0, 1] \times [1, 2] \times [0, \pi]$, we proceed by setting up the bounds for each variable based on the region $E$.

Step-by-Step Solution

1. Identify Integration Bounds:

   • For $x$, $0 \leq x \leq 1$.
   • For $y$, $1 \leq y \leq 2$.
   • For $z$, $0 \leq z \leq \pi$.

2. Set Up the Integral: The integral can be written as: $\int_0^1 \int_1^2 \int_0^{\pi} \frac{e^{-2x} \sin^2(z)}{\sqrt{y}} \, dz \, dy \, dx.$

3. Evaluate the Integral Over $z$: Since $\sin^2(z)$ only depends on $z$, we can integrate with respect to $z$ first: $\int_0^{\pi} \sin^2(z) \, dz = \int_0^{\pi} \frac{1 - \cos(2z)}{2} \, dz.$ Using the identity $\sin^2(z) = \frac{1 - \cos(2z)}{2}$, we can simplify this as: $= \frac{1}{2} \int_0^{\pi} (1 - \cos(2z)) \, dz = \frac{1}{2} \left[ z - \frac{\sin(2z)}{2} \right]_0^{\pi}.$ Evaluating this, we get: $= \frac{1}{2} \left( \pi - 0 \right) = \frac{\pi}{2}.$

4. Evaluate the Integral Over $y$: Now, integrate with respect to $y$: $\int_1^2 \frac{1}{\sqrt{y}} \, dy.$ This integral evaluates as follows: $= \left[ 2 \sqrt{y} \right]_1^2 = 2(\sqrt{2} - 1).$

5. Evaluate the Integral Over $x$: Finally, integrate with respect to $x$: $\int_0^1 e^{-2x} \, dx.$ This evaluates to: $= \left[ -\frac{e^{-2x}}{2} \right]_0^1 = \frac{1}{2} (1 - e^{-2}).$

6. Combine Results: Now, multiply all results together: $\frac{\pi}{2} \cdot 2(\sqrt{2} - 1) \cdot \frac{1}{2} (1 - e^{-2}).$ Simplifying, we get: $= \frac{\pi (\sqrt{2} - 1)(1 - e^{-2})}{2}.$

Final Answer:

$\iiint_E \frac{e^{-2x} \sin^2(z)}{\sqrt{y}} \, dV = \frac{\pi (\sqrt{2} - 1)(1 - e^{-2})}{2}.$

Would you like more details on any specific step?

Here are some related questions you might find useful:

1. How can we use trigonometric identities to simplify integrals involving sine and cosine?
2. What are some techniques for evaluating integrals over rectangular regions?
3. How do exponential functions affect the convergence of integrals?
4. What is the significance of evaluating integrals in different orders of integration?
5. How do we handle integrals that involve square root functions?

Tip: For triple integrals, always set up the bounds for each variable carefully to ensure the correct order of integration.