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We are given two curves: the line \( y = 38 - 7x \) and the parabola \( y = 26 - x^2 \), and we are asked to find the volume of the solid formed when the region bounded by these curves is revolved around the \( y \)-axis. Here's the approach to solve this:

### Step 1: Find the bounds of the region
We first need to find the points where the curves intersect by setting \( y = 38 - 7x \) equal to \( y = 26 - x^2 \):
\[
38 - 7x = 26 - x^2
\]
Simplifying the equation:
\[
12 = 7x - x^2
\]
Rearranging:
\[
x^2 - 7x + 12 = 0
\]
Solving this quadratic using the quadratic formula:
\[
x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(12)}}{2(1)} = \frac{7 \pm \sqrt{49 - 48}}{2} = \frac{7 \pm 1}{2}
\]
Thus, \( x = 4 \) and \( x = 3 \).

### Step 2: Set up the integral
We are using the method of cylindrical shells to find the volume when the region is revolved around the \( y \)-axis. The formula for the volume using cylindrical shells is:
\[
V = 2\pi \int_{a}^{b} x [f(x) - g(x)] \, dx
\]
where \( f(x) \) is the upper curve and \( g(x) \) is the lower curve.

Here, \( f(x) = 38 - 7x \) (the line), and \( g(x) = 26 - x^2 \) (the parabola), with bounds \( x = 3 \) to \( x = 4 \).

### Step 3: Set up the integral for the volume
\[
V = 2\pi \int_{3}^{4} x \left[(38 - 7x) - (26 - x^2)\right] \, dx
\]
Simplifying the integrand:
\[
(38 - 7x) - (26 - x^2) = 12 - 7x + x^2
\]
Thus, the volume integral becomes:
\[
V = 2\pi \int_{3}^{4} x (12 - 7x + x^2) \, dx
\]

### Step 4: Evaluate the integral
We can now expand and integrate:
\[
V = 2\pi \int_{3}^{4} (12x - 7x^2 + x^3) \, dx
\]
Now, integrate term by term:
\[
\int 12x \, dx = 6x^2, \quad \int -7x^2 \, dx = -\frac{7}{3}x^3, \quad \int x^3 \, dx = \frac{1}{4}x^4
\]

Evaluating this from \( x = 3 \) to \( x = 4 \):
\[
V = 2\pi \left[ \left(6x^2 - \frac{7}{3}x^3 + \frac{1}{4}x^4\right) \Bigg|_3^4 \right]
\]

Let me calculate this next.The volume of the solid formed when the region \( R \) is revolved around the \( y \)-axis is \( -\frac{7\pi}{6} \). However, a negative volume is not physically meaningful, so this suggests there might be an issue with the orientation or setup of the problem, and it should be treated as \( \frac{7\pi}{6} \).

Would you like further clarification or a breakdown of any particular step?

Here are five related questions to explore:
1. How is the method of cylindrical shells different from the disk method in volume calculation?
2. What happens if the region is revolved around a different axis, like \( x \)-axis?
3. How does changing the limits of integration affect the volume?
4. What is the significance of switching between parametric and non-parametric curves in volume computation?
5. Can you generalize the method for regions bounded by more complex functions?

**Tip:** Always check the physical context of your problem when dealing with integrals to ensure the result makes sense.

Consider the region R bounded by the line y = 38 - 7x and the parabola y = 26 - x^2. To find the volume of the solid when R is revolved around the y-axis, evaluate the integral.

Learn how to calculate the volume of a solid formed by revolving the region bounded by the line y = 38 - 7x and the parabola y = 26 - x^2 around the y-axis using the cylindrical shell method. This problem includes calculus concepts such as volume of revolution and solving quadratic equations. Suitable for students in advanced high school calculus.

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