Find the volume of the solid obtained by rotating the region bounded by the given curves about the line x=-5: y=x^2, x=y^2 

To find the volume of the solid formed by rotating the region bounded by the curves \( y = x^2 \) and \( x = y^2 \) about the line \( x = -5 \), we will use the **method of cylindrical shells**. This is suitable because we are rotating around a vertical line (i.e., \( x = -5 \)).

### Step 1: Identify the Region

The curves given are:

- \( y = x^2 \) (a parabola opening upwards),
- \( x = y^2 \) (a parabola opening sideways).

We first need to determine where these two curves intersect by solving the system:
\[
y = x^2 \quad \text{and} \quad x = y^2.
\]

Substitute \( y = x^2 \) into \( x = y^2 \):
\[
x = (x^2)^2 = x^4.
\]
So, \( x^4 - x = 0 \), which gives:
\[
x(x^3 - 1) = 0.
\]
Thus, \( x = 0 \) or \( x = 1 \).

- For \( x = 0 \), \( y = 0 \).
- For \( x = 1 \), \( y = 1 \).

Therefore, the curves intersect at the points \( (0, 0) \) and \( (1, 1) \).

### Step 2: Set up the Formula for Cylindrical Shells

The volume \( V \) of the solid of revolution using cylindrical shells is given by:
\[
V = 2\pi \int_{a}^{b} \text{(radius)} \times \text{(height)} \, dx.
\]

Here:
- The **radius** of a shell is the distance from the line \( x = -5 \) to a point on the curve, which is \( |x - (-5)| = x + 5 \).
- The **height** of the shell is the vertical distance between the curves, i.e., the difference in \( y \)-values between the two curves.

- For \( y = x^2 \), the height at any \( x \) is \( y_{\text{upper}} - y_{\text{lower}} = x^2 - (-\sqrt{x}) = x^2\, max width of

Find the volume of the solid obtained by rotating the region bounded by the given curves about the line x=-5: y=x^2, x=y^2

This problem explores how to find the volume of a solid formed by rotating the region bounded by y=x^2 and x=y^2 around the line x=-5. Using the cylindrical shells method, we calculate the volume step by step, covering concepts from calculus and solids of revolution. Suitable for Grades 11-12 and early university calculus 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