Define Q as the region bounded by the functions f(x)=x^2/3 and g(x)=2x in the first quadrant between y=2 and y=3. 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 \( Q \) bounded by \( f(x) = \frac{x^2}{3} \) and \( g(x) = 2x \) around the **y-axis**, we'll again use the **washer method**.

### Steps:

1. **Express the curves in terms of \( x \):**
   We need to express both \( f(x) = \frac{x^2}{3} \) and \( g(x) = 2x \) as functions of \( y \).
   
   - From \( y = \frac{x^2}{3} \), solve for \( x \):
     \[
     y = \frac{x^2}{3} \quad \Rightarrow \quad x = \sqrt{3y}
     \]
   
   - From \( y = 2x \), solve for \( x \):
     \[
     y = 2x \quad \Rightarrow \quad x = \frac{y}{2}
     \]

2. **Determine the limits of integration:**
   We are given that the region is bounded between \( y = 2 \) and \( y = 3 \).

3. **Set up the washer method formula:**
   The volume \( V \) of the solid of revolution using the washer method is given by:
   \[
   V = \pi \int_{y_1}^{y_2} \left( R_{\text{outer}}(y)^2 - R_{\text{inner}}(y)^2 \right) dy
   \]
   Where:
   - \( R_{\text{outer}}(y) \) corresponds to the curve \( g(x) = 2x \) or \( x = \

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

Find the exact volume of the solid formed by rotating the region between f(x)=x^2/3 and g(x)=2x in the first quadrant around the y-axis using the washer method. This problem covers integral calculus and function transformations, suitable for Grades 11-12 and college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation