Use cylindrical coordinates.
Evaluate 
 

E

x2 + y2
 dV,
 where E is the region that lies inside the cylinder 
x2 + y2 = 9
 and between the planes 
z = −4
 and 
z = −1.

To evaluate the integral of \( x^2 + y^2 \) over the region \( E \), where \( E \) is defined by the cylinder \( x^2 + y^2 = 9 \) and the planes \( z = -4 \) and \( z = -1 \), we will switch to cylindrical coordinates.

### Step 1: Cylindrical coordinates transformation
In cylindrical coordinates:
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)
- \( x^2 + y^2 = r^2 \)
- \( dV = r \, dz \, dr \, d\theta \)

Thus, the integral becomes:
\[
\int_E (x^2 + y^2) \, dV = \int_E r^2 \, dV
\]

### Step 2: Set the limits of integration
- The cylinder \( x^2 + y^2 = 9 \) corresponds to \( r = 3 \), so \( r \) will range from \( 0 \) to \( 3 \).
- The angle \( \theta \) in cylindrical coordinates sweeps from \( 0 \) to \( 2\pi \).
- The height is bounded by the planes \( z = -4 \) and \( z = -1 \).

Thus, the limits of integration are:
- \( r: 0 \leq r \leq 3 \)
- \( \theta: 0 \leq \theta \leq 2\pi \)
- \( z: -4 \leq z \leq -1 \)

### Step 3: Set up the triple integral
The integral becomes:
\[
\int_0^{2\pi} \int_0^3 \int_{-4}^{-1} r^2 \cdot r \, dz \, dr \, d\theta = \int_0^{2\pi} \int_0^3 \int_{-4}^{-1} r^3 \, dz \, dr \, d\theta
\]

### Step 4: Evaluate the integrals
- First, integrate with respect to \( z \):
\[
\int_{-4}^{-1} r^3 \, dz = r^3 (z) \bigg|_{-4}^{-1} = r^3(-1 - (-4)) = 3r^3
\]

Now the integral is:
\[
\int_0^{2\pi} \int_0^3 3r^3 \, dr \, d\theta
\]

- Next, integrate with respect to \( r \):
\[
\int_0^3 3r^3 \, dr = 3 \cdot \left( \frac{r^4}{4} \right) \bigg|_0^3 = 3 \cdot \frac{3^4}{4} = 3 \cdot \frac{81}{4} = \frac{243}{4}
\]

Now the integral becomes:
\[
\int_0^{2\pi} \frac{243}{4} \, d\theta
\]

- Finally, integrate with respect to \( \theta \):
\[
\int_0^{2\pi} \frac{243}{4} \, d\theta = \frac{243}{4} \cdot 2\pi = \frac{243 \cdot 2\pi}{4} = \frac{243\pi}{2}
\]

### Final Answer:
The value of the integral is:
\[
\boxed{\frac{243\pi}{2}}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do cylindrical coordinates relate to Cartesian coordinates?
2. How do you compute the volume of a cylinder in cylindrical coordinates?
3. Can you describe the integration limits in spherical coordinates for a similar problem?
4. How would this change if the cylinder radius were different?
5. How can this be applied in evaluating moments of inertia for cylindrical objects?

### Tip:
Always ensure that when switching coordinate systems (e.g., to cylindrical coordinates), you correctly transform both the variables and the volume element \( dV \).

Use cylindrical coordinates. Evaluate ∫_E (x^2 + y^2) dV, where E is the region that lies inside the cylinder x^2 + y^2 = 9 and between the planes z = −4 and z = −1.

Learn how to evaluate the integral ∫_E (x^2 + y^2) dV using cylindrical coordinates. This detailed step-by-step solution explains the transformation process, triple integration, and proper limits of integration for a cylinder with radius 3, between z = −4 and z = −1. Suitable for Calculus III students.

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