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Part 1
Use the shell method to find the volume of the solid generated by revolving the region bounded by the line yequals6xplus7 and the parabola yequalsxsquared about the following lines.
a. The line xequals7
b. The line xequalsminus1
c. The​ x-axis
d. The line yequals49


To solve these problems using the **shell method**, we will need to calculate the volume of the solid generated by revolving the region bounded by the given line \( y = 6x + 7 \) and the parabola \( y = x^2 \) around the different lines specified.

### General formula for the Shell Method:
The formula for the volume of a solid generated by revolving a region around a vertical line \( x = c \) is given by:

\[
V = 2\pi \int_{a}^{b} ( \text{radius} ) \times ( \text{height} ) \, dx
\]

- **Radius**: The distance from the shell to the axis of rotation.
- **Height**: The difference in the \( y \)-values of the two curves at each point.

Let’s go step by step for each part:

### a. Revolving around the line \( x = 7 \)

1. **Find the limits of integration**: 
   The curves \( y = 6x + 7 \) and \( y = x^2 \) intersect where:
   \[
   6x + 7 = x^2
   \]
   Rearranging:
   \[
   x^2 - 6x - 7 = 0
   \]
   Solving this quadratic equation:
   \[
   x = \frac{6 \pm \sqrt{(-6)^2 - 4(1)(-7)}}{2(1)} = \frac{6 \pm \sqrt{36 + 28}}{2} = \frac{6 \pm \sqrt{64}}{2} = \frac{6 \pm 8}{2}
   \]
   So the roots are \( x = 7 \) and \( x = -1 \). Therefore, the limits of integration are \( x = -1 \) to \( x = 7 \).

2. **Calculate the radius and height**:
   - The **radius** is the distance from \( x = 7 \) to the point \( x \), which is \( 7 - x \).
   - The **height** is the difference between the \( y \)-values of the two curves at a given \( x \), so it is \( (6x + 7) - x^2 \).

3. **Set up the integral**:
   The volume is:
   \[
   V = 2\pi \int_{-1}^{7} (7 - x) \times \left( (6x + 7) - x^2 \right) dx
   \]

### b. Revolving around the line \( x = -1 \)

1. **Limits of integration**: 
   As in part (a), the limits of integration are from \( x = -1 \) to \( x = 7 \).

2. **Radius and height**:
   - The **radius** is the distance from \( x = -1 \) to the point \( x \), which is \( x - (-1) = x + 1 \).
   - The **height** is still \( (6x + 7) - x^2 \).

3. **Set up the integral**:
   The volume is:
   \[
   V = 2\pi \int_{-1}^{7} (x + 1) \times \left( (6x + 7) - x^2 \right) dx
   \]

### c. Revolving around the x-axis

1. **Limits of integration**: 
   The limits are from \( x = -1 \) to \( x = 7 \) again, since these are the intersection points.

2. **Radius and height**:
   - The **radius** is simply \( y = (6x + 7) - x^2 \), as we are rotating about the x-axis.
   - The **height** is the same expression for \( y \).

3. **Set up the integral**:
   The volume is:
   \[
   V = 2\pi \int_{-1}^{7} x \times \left( (6x + 7) - x^2 \right) dx
   \]

### d. Revolving around the line \( y = 49 \)

1. **Limits of integration**: 
   The limits are again from \( x = -1 \) to \( x = 7 \).

2. **Radius and height**:
   - The **radius** is the distance from \( y = 49 \) to the curve at each \( x \). For the line \( y = 6x + 7 \), the distance is \( 49 - (6x + 7) = 42 - 6x \). For the parabola \( y = x^2 \), the distance is \( 49 - x^2 \).
   - The **height** is the difference between the \( y \)-values of the two curves at each \( x \), so it is \( (6x + 7) - x^2 \).

3. **Set up the integral**:
   The volume is:
   \[
   V = 2\pi \int_{-1}^{7} \left( 49 - (6x + 7) \right) \times \left( (6x + 7) - x^2 \right) dx
   \]

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Let me know if you'd like to see any of these integrals worked out or if you need further clarification on any of the steps!

Use the shell method to find the volume of the solid generated by revolving the region bounded by the line y=6x+7 and the parabola y=x^2 about the following lines.
a. The line x=7
b. The line x=-1
c. The x-axis
d. The line y=49

Learn how to calculate the volume of solids using the shell method, with detailed steps for revolving the region bounded by the line y=6x+7 and the parabola y=x^2 around various axes. This problem covers key concepts in calculus such as the shell method, volumes of revolution, and solving integrals. Suitable for high school students in Grades 11-12.

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