Use the shell method to find the volume of the solid generated by revolving the region bounded by the line yequalsxplus12 and the parabola yequalsxsquared about y=0

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 line \( y = 0 \) (the x-axis) using the **shell method**, follow these steps:

### Step 1: Determine the points of intersection
The two curves are:

1. \( y = x^2 \) (the parabola)
2. \( y = x + 12 \) (the line)

To find the points of intersection, set the two equations equal to each other:

\[
x^2 = x + 12
\]

Rearrange the equation:

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

Factor the quadratic equation:

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

So, the solutions are \( x = 4 \) and \( x = -3 \).

Thus, the region is bounded between \( x = -3 \) and \( x = 4 \).

### Step 2: Set up the shell method formula
The formula for the volume of a solid using the shell method when revolving around the x-axis (horizontal axis) is:

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

Here:
- The **radius** is the distance from the x-axis, which is \( y \) (the height of each horizontal strip). Since we're revolving around the x-axis, this will be simply \( y \).
- The **height** is the difference between the two curves \( (x + 12) - (x^2) \).

So:

\[
V = 2\pi \int_{-3}^{4} y (\text{height}) \, dy
\]

Substitute the bounds and

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 y = 0

Explore the shell method to compute the volume of a solid formed by revolving the region between the line y = x + 12 and the parabola y = x^2 around the x-axis. Learn step-by-step how to find points of intersection, apply the shell method formula, and calculate the volume. Ideal for college-level calculus students.

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