Use the shell method to find the volume of the solid generated by revolving the region bounded by the line y=x+12 and the parabola y=x^2 about the x-axis

To find the volume of the solid generated by revolving the region bounded by the line \( y = x + 12 \) and the parabola \( y = x^2 \) about the x-axis using the **shell method**, we can proceed step by step:

### Step 1: Set up the shell method formula
For the shell method, the formula for the volume is:

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

- The **radius** is the distance from the axis of rotation (the x-axis) to the height of the shell, which is simply \( y \) (since we revolve around the x-axis).
- The **height** is the difference between the upper curve \( y = x + 12 \) and the lower curve \( y = x^2 \), i.e., \( (x + 12) - x^2 \).

### Step 2: Determine the bounds of integration
To find the limits of integration, we need to find the points where the line and the parabola intersect, which occur when:

\[
x + 12 = x^2
\]

Rearranging this equation:

\[
x^2 - x - 12 = 0
\]

Factoring the quadratic:

\[
(x - 4)(x + 3) = 0
\]

So, the points of intersection are \( x = 4 \) and \( x = -3 \). Therefore, the limits of integration are from \( x = -3 \) to \( x = 4 \).

### Step 3: Apply the shell method
Now we can set up the integral for the volume:

\[
V = 2\pi \int_{-3}^{4} y \left[ (x + 12) - x^2 \right] \, dx
\]

Since the radius \( y \) is simply \( y = x \) and height is \( (x + 12) - x^2 X

Use the shell method to find the volume of the solid generated by revolving the region bounded by the line y=x+12 and the parabola y=x^2 about the x-axis.

Learn how to calculate the volume of a solid generated by revolving the region between the line y = x + 12 and the parabola y = x^2 about the x-axis using the shell method. This solution covers calculus concepts, intersection of curves, and integral calculations. Suitable for college-level calculus students.

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