Find The Volume Of The Solid Obtained By Rotating The Region Bounded By $x = -3 + Y^2$ And $x = 2y$ About The Line $x = -7$. 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 bounded by two curves about a given 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 the curves $x = -3 + y^2$ and $x = 2y$ about the line $x = -7$.

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 given axis. This method is based on the concept of the washer, which is a circular ring with an inner radius and an outer radius. The volume of the solid is then calculated by summing up the volumes of the washers.

Step 1: Find the Points of Intersection

To find the points of intersection, we need to set the two equations equal to each other and solve for $y$. We have:

$$-3 + y^2 = 2y$$

Rearranging the equation, we get:

$$y^2 - 2y - 3 = 0$$

Solving the quadratic equation, we get:

$$y = \frac{2 \pm \sqrt{4 + 12}}{2}$$

$$y = \frac{2 \pm \sqrt{16}}{2}$$

$$y = \frac{2 \pm 4}{2}$$

$$y = 3 \text{ or } y = -1$$

Step 2: Find the Volume of the Solid

To find the volume of the solid, we need to use the method of washers. We will use the formula:

$$V = \pi \int_{a}^{b} (R^2 - r^2) dy$$

where $R$ is the outer radius, $r$ is the inner radius, and $a$ and $b$ are the limits of integration.

In this case, the outer radius is $2y$ and the inner radius is $-3 + y^2$. The limits of integration are $y = -1$ and $y = 3$.

Step 3: Evaluate the Integral

Evaluating the integral, we get:

$$V = \pi \int_{-1}^{3} ((2y)^2 - (-3 + y^2)^2) dy$$

Expanding the integrand, we get:

$$V = \pi \int_{-1}^{3} (4y^2 - 9 + 6y^2 - y^4) dy$$

Simplifying the integrand, we get:

$$V = \pi \int_{-1}^{3} (-y^4 + 10y^2 - 9) dy$$

Evaluating the integral, we get:

$$V = \pi \left[-\frac{y^5}{5} + \frac{10y^3}{3} - 9y\right]_{-1}^{3}$$

Simplifying the expression, we get:

$$V = \pi \left[-\frac{243}{5} + \frac{540}{3} - 27 + \frac{1}{5} - \frac{10}{3} + 9\right]$$

Simplifying further, we get:

$$V = \pi \left[-\frac{243}{5} + 180 - 27 + \frac{1}{5} - \frac{10}{3} + 9\right]$$

Simplifying even further, we get:

$$V = \pi \left[-\frac{243}{5} + \frac{180}{1} - \frac{27}{1} + \frac{1}{5} - \frac{10}{3} + \frac{9}{1}\right]$$

Simplifying even further, we get:

$$V = \pi \left[-\frac{243}{5} + \frac{900}{5} - \frac{135}{5} + \frac{1}{5} - \frac{50}{15} + \frac{45}{5}\right]$$

Simplifying even further, we get:

$$V = \pi \left[\frac{613}{5}\right]$$

Simplifying even further, we get:

$$V = \frac{613\pi}{5}$$

Step 4: Round to the Nearest Thousandth

Rounding the volume to the nearest thousandth, we get:

$$V \approx 385.76$$

Conclusion

In this article, we used the method of washers to find the volume of the solid obtained by rotating the region bounded by the curves $x = -3 + y^2$ and $x = 2y$ about the line $x = -7$. We found that the volume of the solid is approximately $385.76$ cubic units.

References

• [1] "Calculus" by Michael Spivak
• [2] "Calculus: Early Transcendentals" by James Stewart
• [3] "Calculus: Single Variable" by David Guichard

Note

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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 given axis. This method is based on the concept of the washer, which is a circular ring with an inner radius and an outer radius. The volume of the solid is then calculated by summing up the volumes of the washers.

Q: How do I find the points of intersection between two curves?

A: To find the points of intersection, you need to set the two equations equal to each other and solve for $y$. This will give you the $y$-coordinates of the points of intersection.

Q: What is the formula for finding the volume of a solid using the method of washers?

A: The formula for finding the volume of a solid using the method of washers is:

$$V = \pi \int_{a}^{b} (R^2 - r^2) dy$$

where $R$ is the outer radius, $r$ is the inner radius, and $a$ and $b$ are the limits of integration.

Q: How do I evaluate the integral in the formula for finding the volume of a solid using the method of washers?

A: To evaluate the integral, you need to follow the order of operations (PEMDAS):

1. Evaluate the expressions inside the parentheses.
2. Evaluate the exponentiation.
3. Multiply the terms.
4. Add or subtract the terms.

Q: What is the significance of the limits of integration in the formula for finding the volume of a solid using the method of washers?

A: The limits of integration, $a$ and $b$, represent the $y$-coordinates of the points of intersection between the two curves. These limits determine the region of integration, which is the region bounded by the two curves.

Q: Can I use the method of washers to find the volume of a solid obtained by rotating a region about a given axis that is not the x-axis?

A: Yes, you can use the method of washers to find the volume of a solid obtained by rotating a region about a given axis that is not the x-axis. However, you will need to modify the formula for finding the volume of a solid using the method of washers to account for the new axis of rotation.

Q: What are some common applications of the method of washers?

A: The method of washers has numerous applications in various fields, including:

• Physics: to find the volume of a solid obtained by rotating a region about a given axis.
• Engineering: to design and optimize systems that involve rotation, such as gears and pulleys.
• Economics: to model and analyze economic systems that involve rotation, such as the rotation of a company's assets.

Q: What are some common mistakes to avoid when using the method of washers?

A: Some common mistakes to avoid when using the method of washers include:

• Failing to identify the points of intersection between the two curves.
• Failing to evaluate the integral correctly.
• Failing to account for the axis of rotation.
• Failing to round the final answer to the correct number of decimal places.

Q: How can I practice using the method of washers?

A: You can practice using the method of washers by working through examples and exercises in your textbook or online resources. You can also try using the method of washers to solve real-world problems, such as designing a system that involves rotation.