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
 .

The problem involves finding the volume of a solid of revolution formed by rotating the region \( Q \) around the \( y \)-axis. Let's break down the steps and calculate the volume.

### Step 1: Define the region \( Q \)
- The curve is \( f(x) = x \).
- The boundaries are the vertical lines \( x = \frac{1}{2} \) and \( y = 3 \) and \( y = 6 \).

First, we rewrite the equation \( f(x) = x \) in terms of \( y \):
\[
x = y.
\]
This equation will be helpful when rotating around the \( y \)-axis.

### Step 2: Set up the integral using the shell method
Since the region is rotated around the \( y \)-axis, we will use the shell method. The formula for the volume of a solid of revolution using the shell method is:
\[
V = 2\pi \int_{y_1}^{y_2} ( \text{radius of shell}) \cdot (\text{height of shell}) \, dy.
\]

In our case:
- The radius of the shell is \( x = y \), as the region is rotated around the \( y \)-axis.
- The height of the shell is the distance between \( y = 1/2 \) and \( x = 0 \), which is \( \frac{1}{2} \).

So the volume is given by:
\[
V = 2\pta

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.

This problem involves finding the volume of a solid of revolution by rotating the region bounded by f(x) = x and x = 1/2 between y = 3 and y = 6 around the y-axis. Learn how to solve this problem using the shell method and key calculus concepts.

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

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation