Q4. Find the volume of the solid bounded by x=7-y^2-z^2 and x=3.

To find the volume of the solid bounded by the surfaces \( x = 7 - y^2 - z^2 \) and \( x = 3 \), we can set up a triple integral.

### Step 1: Set Up the Equations
- The equation \( x = 7 - y^2 - z^2 \) represents a paraboloid opening towards the negative \( x \)-axis.
- The equation \( x = 3 \) represents a plane perpendicular to the \( x \)-axis.

### Step 2: Determine the Region of Integration
The region of integration is the solid between these two surfaces. 

1. **Intersection of the surfaces**:
   Set \( 7 - y^2 - z^2 = 3 \) to find the boundary in the \( yz \)-plane:
   \[
   4 = y^2 + z^2
   \]
   This is a circle in the \( yz \)-plane with a radius \( r = 2 \).

2. **Limits for \( y \) and \( z \)**:
   \[
   y^2 + z^2 \leq 4
   \]
   This implies \( y \) and \( z \) are bounded within the circle of radius 2:
   \[
   -2 \leq y \leq 2, \quad -\sqrt{4 - y^2} \leq z \leq \sqrt{4 - y^2}
   \]

3. **Limits for \( x \)**:
   For a fixed \( y \) and \( z \),
   \[
   3 \leq x \leq 7 - y^2 - z^2
   \]

### Step 3: Set Up the Integral
The volume \( V \) can be calculated by integrating over the region \( D \):
\[
V = \int_{-2}^{2} \int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}} \int_{3}^{7-y^2-z^2} dx \, dz \, dy
\]

### Step 4: Evaluate the Integral
Start by integrating with respect to \( x \):
\[
\int_{3}^{7-y^2-z^2} dx = (7 - y^2 - z^2) - 3 = 4 - y^2 - z^2
\]
So the volume integral becomes:
\[
V = \int_{-2}^{2} \int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}} (4 - y^2 - z^2) \, dz \, dy
\]

### Step 5: Simplify and Integrate
Convert to polar coordinates:
- Let \( y = r \cos\theta \) and \( z = r \sin\theta \)
- Then, \( y^2 + z^2 = r^2 \) and the Jacobian of the transformation is \( r \).

Now, the volume integral is:
\[
V = \int_{0}^{2\pi} \int_{0}^{2} (4 - r^2) r \, dr \, d\theta
\]

First, integrate with respect to \( r \):
\[
\int_{0}^{2} (4r - r^3) \, dr = \left[2r^2 - \frac{r^4}{4}\right]_{0}^{2} = \left[8 - 4\right] = 4
\]

Finally, integrate with respect to \( \theta \):
\[
V = \int_{0}^{2\pi} 4 \, d\theta = 4 \times 2\pi = 8\pi
\]

### Final Answer:
The volume of the solid is \( \boxed{8\pi} \).

---

Would you like more details on this process, or do you have any other questions?

### Related Questions:
1. How do you find the surface area of the same solid?
2. What is the centroid of the given solid?
3. How would you calculate the volume if the paraboloid had a different equation?
4. Can the method of cylindrical shells be used to find this volume?
5. How does changing the plane \( x = 3 \) affect the volume?
6. What happens to the volume if the paraboloid is rotated about the \( x \)-axis?
7. How does one convert a double integral in Cartesian coordinates to polar coordinates in general?
8. What is the importance of the Jacobian when converting to polar coordinates?

### Tip:
When converting from Cartesian to polar coordinates, remember that the radius \( r \) represents the distance from the origin, and the Jacobian determinant helps account for the area scaling in the new coordinate system.

Learn how to calculate the volume of a solid bounded by the surfaces x=7-y^2-z^2 and x=3 using triple integrals. This problem involves advanced mathematical concepts such as triple integrals and volume calculations. Suitable for advanced college-level students.

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