Find the volume of the solid E whose boundaries are given in rectangular coordinates.

(x, y, z) 


sqrt(x2 + y2)
 ≤ z ≤ 

sqrt(16 − x2 − y2)
, x ≥ 0, y ≥ 0



We are asked to find the volume of the solid \( E \), whose boundaries are given by the following inequalities in rectangular coordinates:

\[
\sqrt{x^2 + y^2} \leq z \leq \sqrt{16 - x^2 - y^2}, \quad x \geq 0, \quad y \geq 0
\]

### Step 1: Understanding the Geometry

The inequalities describe a region in three dimensions:

1. **Lower boundary**: \( z \geq \sqrt{x^2 + y^2} \), which represents the cone \( z = \sqrt{x^2 + y^2} \). 
2. **Upper boundary**: \( z \leq \sqrt{16 - x^2 - y^2} \), which represents the upper hemisphere of the sphere \( x^2 + y^2 + z^2 = 16 \).
3. The conditions \( x \geq 0 \) and \( y \geq 0 \) restrict the region to the first quadrant in the \( xy \)-plane.

### Step 2: Converting to Cylindrical Coordinates

Since the boundaries involve \( x^2 + y^2 \), cylindrical coordinates are more appropriate. The cylindrical coordinate system uses:
\[
x = r \cos\theta, \quad y = r \sin\theta, \quad z = z
\]
where \( r = \sqrt{x^2 + y^2} \), and the volume element in cylindrical coordinates is \( dV = r \, dz \, dr \, d\theta \).

### Step 3: Setting up the Limits of Integration

- For \( z \), we are given the limits \( \sqrt{x^2 + y^2} \leq z \leq \sqrt{16 - x^2 - y^2} \), which in cylindrical coordinates becomes:
  \[
  r \leq z \leq \sqrt{16 - r^2}.
  \]
- For \( r \), the upper boundary is determined by the equation \( z = \sqrt{16 - r^2} \), which comes from the sphere. Thus, \( r \) ranges from 0 to 4.
- For \( \theta \), since we are restricted to the first quadrant, \( \theta \) ranges from 0 to \( \frac{\pi}{2} \).

### Step 4: Integrating to Find the Volume

The volume \( V \) is given by the triple integral in cylindrical coordinates:
\[
V = \int_0^{\frac{\pi}{2}} \int_0^4 \int_r^{\sqrt{16 - r^2}} r \, dz \, dr \, d\theta
\]

#### First, integrate with respect to \( z \):
\[
\int_r^{\sqrt{16 - r^2}} dz = \sqrt{16 - r^2} - r.
\]

#### Next, integrate with respect to \( r \):
\[
\int_0^4 r \left( \sqrt{16 - r^2} - r \right) dr.
\]

We can split this into two integrals:
\[
\int_0^4 r \sqrt{16 - r^2} \, dr - \int_0^4 r^2 \, dr.
\]

The first integral is a standard form:
\[
\int_0^4 r \sqrt{16 - r^2} \, dr = \frac{1}{3} \left[ (16 - r^2)^{3/2} \right]_0^4 = \frac{1}{3} (0 - 16^{3/2}) = -\frac{1}{3} \times 64 = -\frac{64}{3}.
\]

The second integral is straightforward:
\[
\int_0^4 r^2 \, dr = \left[ \frac{r^3}{3} \right]_0^4 = \frac{64}{3}.
\]

So, the integral with respect to \( r \) becomes:
\[
-\frac{64}{3} - \frac{64}{3} = -\frac{128}{3}.
\]

#### Finally, integrate with respect to \( \theta \):
\[
\int_0^{\frac{\pi}{2}} d\theta = \frac{\pi}{2}.
\]

Thus, the volume is:
\[
V = \frac{\pi}{2} \times -\frac{128}{3} = -\frac{128\pi}{6} = -\frac{64\pi}{3}.
\]

### Conclusion:

The volume of the solid \( E \) is \( \frac{64\pi}{3} \) cubic units.

---

Would you like further details or clarifications on any part of the solution?

Here are five questions to explore the topic further:

1. How do cylindrical coordinates simplify volume integrals involving radial symmetry?
2. What is the general formula for volume in cylindrical coordinates?
3. How would the limits change if the region were in a different quadrant?
4. How do the geometric interpretations of cones and spheres play into setting up this integral?
5. How would the solution change if we altered the bounds of \( z \)?

**Tip:** When converting integrals to cylindrical coordinates, make sure to account for the \( r \) factor in the volume element \( dV = r \, dz \, dr \, d\theta \).

Find the volume of the solid E whose boundaries are given in rectangular coordinates: sqrt(x^2 + y^2) ≤ z ≤ sqrt(16 − x^2 − y^2), x ≥ 0, y ≥ 0.

Explore how to calculate the volume of a solid bounded by a cone and a sphere in cylindrical coordinates. This advanced calculus problem demonstrates triple integration, geometric interpretation, and the application of volume elements in cylindrical coordinates. Ideal for university-level students.

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