Find the volume of the solid obtained by rotating the region bounded by the given curves about the line x=10. x=y^2, x=1

We are tasked with finding the volume of the solid obtained by rotating the region bounded by the curves \( x = y^2 \) and \( x = 1 \) about the line \( x = 10 \).

### Step 1: Analyze the region
The curves given are:

- \( x = y^2 \): A parabola that opens to the right.
- \( x = 1 \): A vertical line.

The region of interest is bounded horizontally by \( x = y^2 \) and \( x = 1 \). To determine the vertical limits of integration, solve \( x = 1 \) for \( y \) using \( x = y^2 \):
\[
y^2 = 1 \implies y = -1 \text{ and } y = 1.
\]
Thus, the region is bounded by \( y = -1 \) and \( y = 1 \).

### Step 2: Set up the washer method formula
We will use the washer method since we are rotating the region around \( x = 10 \), which is a vertical line.

The formula for the volume using the washer method is:
\[
V = \pi \int_{y_1}^{y_2} \left[ (R_{\text{outer}}(y))^2 - (R_{\text{inner}}(y))^2 \right] \, dy,
\]
where:
- \( R_{\text{outer}}(y) \) is the distance from the line \( x = 10 \) to the outer curve (which in this case is \( x = 1 \)),
- \( R_{\text{inner}}(y) \) is the distance from the line \( x = 10 \) to the inner curve (which in this case is \( x = y^2 \)).

### Step 3: Calculate the radii
- For the outer radius (the distance from \( x = 10 \) to \( x = 1 \)): 
  \[
  R_{\text{outer}}(y) = 10 - 1 = 9.
  \]
- For the inner radius (the distance from \( x = 10 \) to \( x = y^2 \)):
  \[
  R_{\text{inner}}(y) = 10 - y^2.
  \]

### Step 4: Set up the integral
The volume is thus:
\[
V = \pi \int_{-1}^{1} \left[ 9^2 - (10 - y^2)^2 \right] \, dy.
\]
Simplify the integrand:
\[
9^2 = 81,
\]
\[
(10 - y^2)^2 = (10^2 - 2(10)(y^2) + (y^2)^2) = 100 - 20y^2 + y^4.
\]
So the integrand becomes:
\[
81 - (100 - 20y^2 + y^4) = 81 - 100 + 20y^2 - y^4 = -19 + 20y^2 - y^4.
\]

### Step 5: Compute the integral
Now, integrate term by term:
\[
V = \pi \int_{-1}^{1} \left( -19 + 20y^2 - y^4 \right) \, dy.
\]
Because the integrand contains only even powers of \( y \), we can double the integral from 0 to 1:
\[
V = 2\pi \int_0^1 \left( -19 + 20y^2 - y^4 \right) \, dy.
\]
Now, integrate each term:
\[
\int_0^1 -19 \, dy = -19y \Big|_0^1 = -19,
\]
\[
\int_0^1 20y^2 \, dy = 20 \cdot \frac{y^3}{3} \Big|_0^1 = \frac{20}{3},
\]
\[
\int_0^1 -y^4 \, dy = -\frac{y^5}{5} \Big|_0^1 = -\frac{1}{5}.
\]

### Step 6: Final calculation
Now, summing these results:
\[
V = 2\pi \left( -19 + \frac{20}{3} - \frac{1}{5} \right).
\]
To combine the terms, find a common denominator:
\[
-19 = \frac{-285}{15}, \quad \frac{20}{3} = \frac{100}{15}, \quad \frac{1}{5} = \frac{3}{15},
\]
so:
\[
V = 2\pi \left( \frac{-285 + 100 - 3}{15} \right) = 2\pi \left( \frac{-188}{15} \right).
\]
Thus:
\[
V = \frac{-376\pi}{15}.
\]
Since volume cannot be negative, the absolute value is taken, so the volume is:
\[
V = \frac{376\pi}{15}.
\]

### Final Answer:
The volume of the solid is \( \frac{376\pi}{15} \).

Would you like to explore further details or clarifications on any part of this solution?

Here are 5 related questions you could consider:
1. How would the volume change if the region were rotated about the y-axis?
2. What is the volume if the region is rotated about \( x = -1 \) instead of \( x = 10 \)?
3. Can you compute the centroid of the region bounded by \( x = y^2 \) and \( x = 1 \)?
4. What if the boundary curve were \( x = y^3 \) instead of \( x = y^2 \)?
5. How does the washer method differ from the cylindrical shell method for this problem?

**Tip:** When calculating volumes of solids of revolution, always check if the integrand is simplified correctly to avoid errors during integration.

Calculate the volume of a solid formed by rotating the region bounded by x=y^2 and x=1 about the line x=10 using the washer method. This problem involves definite integration and calculus concepts, ideal for students in Grades 11-12 or college-level calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation