To solve this problem, we are asked to evaluate the triple integral using cylindrical coordinates. Here's the detailed solution approach:

Step 1: Set up the Integral in Cylindrical Coordinates

Given: $\iiint_E 5\sqrt{x^2 + y^2} \, dV$ where $E$ is the region inside the cylinder $x^2 + y^2 = 9$ and between the planes $z = -2$ and $z = 3$.

In cylindrical coordinates, we have:

• $x = r \cos \theta$,
• $y = r \sin \theta$,
• $z = z$,
• $x^2 + y^2 = r^2$, so $\sqrt{x^2 + y^2} = r$.

The volume element $dV$ in cylindrical coordinates is $dV = r \, dr \, d\theta \, dz$.

Step 2: Rewrite the Integral

Substitute $\sqrt{x^2 + y^2} = r$ and $dV = r \, dr \, d\theta \, dz$ into the integral: $\iiint_E 5\sqrt{x^2 + y^2} \, dV = \iiint_E 5r \cdot r \, dr \, d\theta \, dz = \iiint_E 5r^2 \, dr \, d\theta \, dz$

Step 3: Set up the Limits of Integration

For the region $E$:

• $r$ ranges from $0$ to $3$ (since $x^2 + y^2 = 9$ gives $r = 3$),
• $\theta$ ranges from $0$ to $2\pi$ (full circle around the cylinder),
• $z$ ranges from $-2$ to $3$ (between the given planes).

So, the integral becomes: $\int_{z=-2}^{3} \int_{\theta=0}^{2\pi} \int_{r=0}^{3} 5r^2 \, dr \, d\theta \, dz$

Step 4: Evaluate the Integral

1. Integrate with respect to $r$: $\int_{0}^{3} 5r^2 \, dr = 5 \int_{0}^{3} r^2 \, dr = 5 \left[ \frac{r^3}{3} \right]_{0}^{3} = 5 \cdot \frac{27}{3} = 5 \cdot 9 = 45$

2. Integrate with respect to $\theta$: $\int_{0}^{2\pi} 45 \, d\theta = 45 \cdot 2\pi = 90\pi$

3. Integrate with respect to $z$: $\int_{-2}^{3} 90\pi \, dz = 90\pi \cdot \left[ z \right]_{-2}^{3} = 90\pi \cdot (3 + 2) = 90\pi \cdot 5 = 450\pi$

Final Answer

$\iiint_E 5\sqrt{x^2 + y^2} \, dV = 450\pi$

Would you like further details on any specific part of this solution?

Here are five related questions you might find interesting:

1. How do cylindrical coordinates simplify integration for circular regions?
2. Why is $dV = r \, dr \, d\theta \, dz$ in cylindrical coordinates?
3. How would this problem change if we were given spherical coordinates instead?
4. What is the significance of choosing cylindrical coordinates for a cylinder-shaped region?
5. How does the integration process change when the function being integrated is not symmetric?

Tip: For problems involving circular or cylindrical symmetry, cylindrical coordinates often simplify the integration process, reducing complexity and making the setup more intuitive.