We are tasked with finding the volume of the solid obtained by rotating the region in the first quadrant bounded by the following curves around the y-axis:

1. $y = x^7$
2. $y = 1$
3. The y-axis ($x = 0$)

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Step 1: Identify the method

Since we are rotating around the y-axis, the most appropriate method is the shell method or disk method. In this problem, the shell method is simpler.

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Step 2: Shell Method Formula

The shell method for volume around the y-axis is given by:

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

Where:

• Radius = $x$ (since we are rotating around the y-axis)
• Height = $1 - x^7$ (the vertical distance between $y = 1$ and $y = x^7$)

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Step 3: Set up the integral

We are integrating with respect to $x$ from $x = 0$ to $x = 1$ (since $y = x^7 = 1$ at $x = 1$):

$V = 2\pi \int_{0}^{1} x(1 - x^7) \, dx$

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Step 4: Compute the integral

We will expand and integrate:

$V = 2\pi \int_{0}^{1} \left( x - x^8 \right) \, dx$

Integral of each term:

$\int x \, dx = \frac{x^2}{2}, \quad \int x^8 \, dx = \frac{x^9}{9}$

Thus:

$V = 2\pi \left[ \frac{x^2}{2} - \frac{x^9}{9} \right]_{0}^{1}$

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Step 5: Evaluate the integral

Substitute the bounds:

[ V = 2\pi \left( \left[ \frac{1^2}{2} - \frac{1^9}{9} \right] - \left[ \frac{0^2}{2} - \frac{0^9}{9} \right] \right) ]

Simplifying:

$V = 2\pi \left( \frac{1}{2} - \frac{1}{9} \right)$

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Step 6: Simplify the expression

To combine the fractions:

$\frac{1}{2} - \frac{1}{9} = \frac{9 - 2}{18} = \frac{7}{18}$

Thus, the volume is:

$V = 2\pi \cdot \frac{7}{18} = \frac{7\pi}{9}$

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Final Answer:

$V = \frac{7\pi}{9}$

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Let me know if you’d like the detailed explanation for any step! 😊

Here are 5 related questions for further practice:

1. What is the volume of the solid obtained by rotating $y = \sqrt{x}$ around the y-axis from $y = 0$ to $y = 4$?
2. Use the disk method to find the volume of the solid generated by rotating $y = x^3$ about the x-axis from $x = 0$ to $x = 2$.
3. How do you determine when to use the disk method vs. the shell method?
4. What happens if we rotate the same region around the x-axis?
5. How do you find the centroid of the region bounded by $y = x^7$ and $y = 1$?

Tip: The shell method is often more efficient when rotating around the y-axis if the function is given as $y = f(x)$.