Find The Volume Of The Solid Obtained By Rotating The Region Bounded By $x = -2 + Y^2$ And $x = -y$ About The Line $x = -3$. Round To The Nearest Thousandth.

Introduction

In this article, we will explore the concept of finding the volume of a solid obtained by rotating a region about a line. This is a fundamental problem in calculus, and it has numerous applications in various fields such as physics, engineering, and economics. We will use the method of washers to find the volume of the solid obtained by rotating the region bounded by $x = -2 + y^2$ and $x = -y$ about the line $x = -3$.

Understanding the Problem

To begin with, let's understand the problem at hand. We are given two curves, $x = -2 + y^2$ and $x = -y$, and we need to find the volume of the solid obtained by rotating the region bounded by these two curves about the line $x = -3$. This means that we will be rotating the region about a vertical line, and we need to find the volume of the resulting solid.

Visualizing the Region

To visualize the region, let's first find the points of intersection between the two curves. We can do this by setting the two equations equal to each other and solving for $y$. This gives us:

$$-2 + y^2 = -y$$

Simplifying the equation, we get:

$$y^2 + y - 2 = 0$$

Factoring the quadratic equation, we get:

$$(y + 2)(y - 1) = 0$$

This gives us two possible values for $y$: $y = -2$ and $y = 1$. Substituting these values back into one of the original equations, we get the corresponding values of $x$: $x = 2$ and $x = -1$.

Finding the Volume

Now that we have visualized the region, let's find the volume of the solid obtained by rotating the region about the line $x = -3$. To do this, we will use the method of washers. The method of washers is a technique used to find the volume of a solid obtained by rotating a region about a line. It involves finding the area of the region and then multiplying it by the distance between the line and the region.

The area of the region can be found by integrating the difference between the two curves with respect to $y$. This gives us:

$$A(y) = \int_{-2}^{1} (-y + (-2 + y^2)) dy$$

Evaluating the integral, we get:

$$A(y) = \left[-\frac{y^2}{2} - 2y + \frac{y^4}{4}\right]_{-2}^{1}$$

Simplifying the expression, we get:

$$A(y) = \frac{3}{4} + 2$$

The distance between the line $x = -3$ and the region can be found by subtracting the $x$-coordinate of the line from the $x$-coordinate of the region. This gives us:

$$d(y) = (-3) - (-2 + y^2)$$

Simplifying the expression, we get:

$$d(y) = 1 - y^2$$

Now that we have found the area and the distance, we can find the volume of the solid obtained by rotating the region about the line $x = -3$. This can be done by integrating the product of the area and the distance with respect to $y$. This gives us:

$$V = \int_{-2}^{1} A(y) d(y) dy$$

Evaluating the integral, we get:

$$V = \int_{-2}^{1} \left(\frac{3}{4} + 2\right) (1 - y^2) dy$$

Simplifying the expression, we get:

$$V = \left[\frac{3}{4}y - \frac{y^3}{3} + 2y - \frac{2y^3}{3}\right]_{-2}^{1}$$

Evaluating the limits, we get:

$$V = \frac{3}{4} - \frac{1}{3} + 2 - \frac{2}{3}$$

Simplifying the expression, we get:

$$V = \frac{7}{12}$$

Rounding to the Nearest Thousandth

Finally, we need to round the volume to the nearest thousandth. This means that we need to round the decimal part of the volume to the nearest thousandth. In this case, the decimal part of the volume is $\frac{7}{12} = 0.5833...$. Rounding this to the nearest thousandth, we get:

$$V \approx 0.583$$

Therefore, the volume of the solid obtained by rotating the region bounded by $x = -2 + y^2$ and $x = -y$ about the line $x = -3$ is approximately $0.583$ cubic units.

Conclusion

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Q: What is the method of washers?

A: The method of washers is a technique used to find the volume of a solid obtained by rotating a region about a line. It involves finding the area of the region and then multiplying it by the distance between the line and the region.

Q: How do I find the area of the region?

A: To find the area of the region, you need to integrate the difference between the two curves with respect to $y$. This will give you the area of the region.

Q: What is the distance between the line and the region?

A: The distance between the line and the region can be found by subtracting the $x$-coordinate of the line from the $x$-coordinate of the region.

Q: How do I find the volume of the solid obtained by rotating the region about the line?

A: To find the volume of the solid obtained by rotating the region about the line, you need to integrate the product of the area and the distance with respect to $y$. This will give you the volume of the solid.

Q: What is the formula for finding the volume of the solid obtained by rotating the region about the line?

A: The formula for finding the volume of the solid obtained by rotating the region about the line is:

$$V = \int_{a}^{b} A(y) d(y) dy$$

where $A(y)$ is the area of the region and $d(y)$ is the distance between the line and the region.

Q: How do I evaluate the integral to find the volume of the solid?

A: To evaluate the integral, you need to follow the order of operations and simplify the expression. This will give you the volume of the solid.

Q: What is the significance of finding the volume of a solid obtained by rotating a region about a line?

A: Finding the volume of a solid obtained by rotating a region about a line is significant in various fields such as physics, engineering, and economics. It has numerous applications in these fields, including calculating the volume of a container, the amount of material needed for a construction project, and the volume of a liquid in a container.

Q: What are some common mistakes to avoid when finding the volume of a solid obtained by rotating a region about a line?

A: Some common mistakes to avoid when finding the volume of a solid obtained by rotating a region about a line include:

• Not following the order of operations when evaluating the integral
• Not simplifying the expression after evaluating the integral
• Not considering the distance between the line and the region
• Not using the correct formula for finding the volume of the solid

Q: How can I practice finding the volume of a solid obtained by rotating a region about a line?

A: You can practice finding the volume of a solid obtained by rotating a region about a line by working through examples and exercises. You can also use online resources and calculators to help you with the calculations.

Q: What are some real-world applications of finding the volume of a solid obtained by rotating a region about a line?

A: Some real-world applications of finding the volume of a solid obtained by rotating a region about a line include:

• Calculating the volume of a container
• Determining the amount of material needed for a construction project
• Finding the volume of a liquid in a container
• Calculating the volume of a solid object

Q: How can I use the method of washers to find the volume of a solid obtained by rotating a region about a line?

A: You can use the method of washers to find the volume of a solid obtained by rotating a region about a line by following these steps:

1. Find the area of the region
2. Find the distance between the line and the region
3. Integrate the product of the area and the distance with respect to $y$
4. Evaluate the integral to find the volume of the solid

By following these steps, you can use the method of washers to find the volume of a solid obtained by rotating a region about a line.