Define
Q
as the region bounded by the functions
f
(
x
)
=
x
and
x
=
1
2
in the first quadrant between
y
=
3
and
y
=
6
. If
Q
is rotated around the
y
-axis, what is the volume of the resulting solid?

Submit an exact answer in terms of
π
.

To find the volume of the solid formed by rotating the region \( Q \) around the \( y \)-axis, we will use the method of cylindrical shells.

### Step 1: Define the region \( Q \)

The region \( Q \) is bounded by:
- The function \( f(x) = x \), which is \( x = y \) (or \( y = x \) rearranged).
- The vertical line \( x = \frac{1}{2} \).
- The horizontal lines \( y = 3 \) and \( y = 6 \).

The region lies in the first quadrant.

### Step 2: Set up the volume integral using the method of cylindrical shells

We rotate the region around the \( y \)-axis. The method of cylindrical shells is given by the formula:

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

In our case:
- The rotation is around the \( y \)-axis.
- We use vertical strips parallel to the \( x \)-axis.
  
We will integrate with respect to \( y \), so let's express \( x \) in terms of \( y \). From \( f(x) = x \), we have \( x = y \). Thus, our limits are from \( y = 3 \) to \( y = 6 \).

The distance from the \( y \)-axis to the region is \( x \), which is equal to \( y \), and the height of each shell is given by the difference \( \left( 1 - x \right) = 1 - \frac{1}{2} = height

The cylindrical shell volume will be, upon integrating this over the function over that height interval  so our total is given, based on now computing bounds formating with pi and its given so. 

Define Q as the region bounded by the functions f(x) = x and x = 1/2 in the first quadrant between y = 3 and y = 6. If Q is rotated around the y-axis, what is the volume of the resulting solid? Submit an exact answer in terms of π.

Learn how to calculate the volume of a solid formed by rotating the region defined by f(x) = x and x = 1/2 between y = 3 and y = 6 around the y-axis. This problem demonstrates the use of the cylindrical shell method in calculus to solve for the volume of a solid of revolution. Suitable for college-level calculus students.

Math Problem Statement

Define Q as the region bounded by the functions f ( x )

x and x

1 2 in the first quadrant between y

3 and y

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