https://file.aiquickdraw.com/mathbot/file/1729205760793igy8c0x1.jpg

To solve the problem of finding the volume of the solid generated by revolving the region bounded by the graphs of the given equations around the y-axis, we can use the **method of disks** or **washers**. Let's break it down:

### Step 1: Identify the curves and bounds
We are given the following:
1. \( x = 1 + y^2 \)
2. \( y = x - 3 \) or equivalently, \( x = y + 3 \)

We are rotating the region around the **y-axis**. First, let's find the intersection points of the two curves by setting their equations equal to each other.

\[
1 + y^2 = y + 3
\]

Rearranging terms:

\[
y^2 - y - 2 = 0
\]

Factoring the quadratic equation:

\[
(y - 2)(y + 1) = 0
\]

Thus, \( y = 2 \) and \( y = -1 \). These are the limits of integration.

### Step 2: Set up the volume integral
To find the volume when rotating about the y-axis, we use the formula for the volume of a solid of revolution:

\[
V = \pi \int_{a}^{b} \left[ R_{\text{outer}}(y)^2 - R_{\text{inner}}(y)^2 \right] \, dy
\]

In this case:
- The outer radius \( R_{\text{outer}}(y) \) corresponds to the line \( x = y + 3 \), since it's farther from the y-axis.
- The inner radius \( R_{\text{inner}}(y) \) corresponds to the curve \( x = 1 + y^2 \).

Thus, the volume integral becomes:

\[
V = \pi \int_{-1}^{2} \left[ (y + 3)^2 - (1 + y^2)^2 \right] \, dy
\]

### Step 3: Compute the integral
First, expand each term:

\[
(y + 3)^2 = y^2 + 6y + 9
\]
\[
(1 + y^2) = y^2 + 1
\]

Thus, the integral becomes:

\[
V = \pi \int_{-1}^{2} \left[ (y^2 + 6y + 9) - (y^2 + 1) \right] \, dy
\]
Simplifying:

\[
V = \pi \int_{-1}^{2} \left[ 6y + 8 \right] \, dy
\]

### Step 4: Solve the integral
Now, integrate each term:

\[
V = \pi \left[ 3y^2 + 8y \right]_{-1}^{2}
\]

Evaluate at the bounds \( y = 2 \) and \( y = -1 \):

For \( y = 2 \):
\[
3(2)^2 + 8(2) = 3(4) + 16 = 12 + 16 = 28
\]

For \( y = -1 \):
\[
3(-1)^2 + 8(-1) = 3(1) - 8 = 3 - 8 = -5
\]

Thus, the total volume is:

\[
V = \pi \left( 28 - (-5) \right) = \pi(28 + 5) = \pi \times 33 = 33\pi
\]

However, this result doesn't match the choices exactly. Let me recheck.

We probably missed checking if there's a washer effect, so the next steps need to review

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the indicated line: x = 1 + y^2, y = x - 3; about the y-axis.

Learn how to compute the volume of a solid generated by revolving the region between the curves x = 1 + y^2 and y = x - 3 around the y-axis. This calculus problem involves the method of washers and solving quadratic equations, suitable for Calculus I-II 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