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We are asked to find the volume of the region enclosed by the curve \( y = 1.6 - 1|x - 15| \) and the line \( y = 0 \), rotated around the \( y \)-axis, using the method of cylindrical shells.

### Step-by-Step Solution Using Cylindrical Shells:
The cylindrical shells method formula for volume is:

\[
V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx
\]

Where:
- \( x \) is the radius of the shell (the distance from the \( y \)-axis),
- \( f(x) \) is the height of the shell (the function describing the curve),
- The limits \( a \) and \( b \) are the bounds of the region.

### 1. Set up the equation and region
The given function is:
\[
y = 1.6 - |x - 15|
\]
This describes a V-shaped region symmetric around \( x = 15 \) in the first quadrant, with the intercept at \( x = 15 \). The equation simplifies to two linear pieces:
- For \( x \geq 15 \): \( y = 1.6 - (x - 15) = 16 - x \),
- For \( x \leq 15 \): \( y = 1.6 - (x - 15) = you and



Use cylindrical shells to find the volume formed by rotating the region in the first quadrant enclosed by: y = 1.6 - |x - 15| and y = 0 about the y-axis.

Learn how to calculate the volume of a region in the first quadrant enclosed by y = 1.6 - |x - 15| and y = 0, rotated about the y-axis using the method of cylindrical shells. This problem covers key calculus concepts suitable for college-level students.

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