Revolve The Region Bounded By $y=x^2$ And $y=x^3$ About The $y − A X I S -axis − A X I S And Find The Volume. V = [ ? ] V=[?] V = [ ?] Round Your Answer To The Nearest Thousandth.

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Introduction

In this article, we will explore the concept of revolving a region about the $y$-axis and finding the volume of the resulting solid. We will focus on the specific case where the region is bounded by the curves $y=x^2$ and $y=x^3$. Our goal is to find the volume of the solid formed by revolving this region about the $y$-axis.

Background and Notation

To begin, let's establish some notation and background information. We are given two curves, $y=x^2$ and $y=x^3$, which intersect at the origin $(0,0)$. We will be revolving the region bounded by these curves about the $y$-axis, which means that we will be rotating the region around the $y$-axis.

Finding the Volume of the Solid

To find the volume of the solid formed by revolving the region about the $y$-axis, we will use the method of washers. This method involves finding the volume of the solid formed by revolving a region about an axis, and it is based on the concept of the washer method.

The washer method states that the volume of a solid formed by revolving a region about an axis is equal to the volume of the solid formed by revolving the region about the axis, minus the volume of the solid formed by revolving the region about the axis, plus the volume of the solid formed by revolving the region about the axis.

In this case, we will be revolving the region bounded by $y=x^2$ and $y=x^3$ about the $y$-axis. To find the volume of the solid formed by revolving this region, we will need to find the volume of the solid formed by revolving the region about the $y$-axis, minus the volume of the solid formed by revolving the region about the $y$-axis, plus the volume of the solid formed by revolving the region about the $y$-axis.

Calculating the Volume of the Solid

To calculate the volume of the solid formed by revolving the region about the $y$-axis, we will need to find the volume of the solid formed by revolving the region about the $y$-axis, minus the volume of the solid formed by revolving the region about the $y$-axis, plus the volume of the solid formed by revolving the region about the $y$-axis.

The volume of the solid formed by revolving the region about the $y$-axis is given by the formula:

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

where $R$ and $r$ are the outer and inner radii of the washer, respectively, and $a$ and $b$ are the limits of integration.

In this case, the outer radius of the washer is given by $R = x^3$, and the inner radius of the washer is given by $r = x^2$. The limits of integration are $a = 0$ and $b = 1$.

Substituting these values into the formula, we get:

$$V = \pi \int_{0}^{1} (x^6 - x^4) dx$$

Evaluating the Integral

To evaluate the integral, we will use the power rule of integration, which states that:

$$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$

Applying this rule to the integral, we get:

$$V = \pi \left[ \frac{x^7}{7} - \frac{x^5}{5} \right]_{0}^{1}$$

Simplifying the Expression

To simplify the expression, we will evaluate the limits of integration.

$$V = \pi \left[ \frac{1^7}{7} - \frac{1^5}{5} \right] - \pi \left[ \frac{0^7}{7} - \frac{0^5}{5} \right]$$

$$V = \pi \left[ \frac{1}{7} - \frac{1}{5} \right]$$

Simplifying the Fraction

To simplify the fraction, we will find a common denominator, which is 35.

$$V = \pi \left[ \frac{5}{35} - \frac{7}{35} \right]$$

$$V = \pi \left[ \frac{-2}{35} \right]$$

Multiplying by Pi

To multiply by pi, we will multiply the fraction by pi.

$$V = \pi \left[ \frac{-2}{35} \right]$$

$$V = \frac{-2\pi}{35}$$

Rounding to the Nearest Thousandth

To round to the nearest thousandth, we will round the value of pi to the nearest thousandth.

$$V = \frac{-2(3.14159)}{35}$$

$$V = \frac{-6.28318}{35}$$

$$V = -0.179$$

Conclusion

In this article, we have explored the concept of revolving a region about the $y$-axis and finding the volume of the resulting solid. We have focused on the specific case where the region is bounded by the curves $y=x^2$ and $y=x^3$. Our goal was to find the volume of the solid formed by revolving this region about the $y$-axis.

Using the method of washers, we have found that the volume of the solid formed by revolving the region about the $y$-axis is given by the formula:

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

where $R$ and $r$ are the outer and inner radii of the washer, respectively, and $a$ and $b$ are the limits of integration.

We have evaluated the integral and simplified the expression to find that the volume of the solid formed by revolving the region about the $y$-axis is:

$$V = \frac{-2\pi}{35}$$

Rounding to the nearest thousandth, we get:

$$V = -0.179$$

This is the final answer to the problem.

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Introduction

In our previous article, we explored the concept of revolving a region about the $y$-axis and finding the volume of the resulting solid. We focused on the specific case where the region is bounded by the curves $y=x^2$ and $y=x^3$. In this article, we will answer some of the most frequently asked questions about this topic.

Q: What is the method of washers?

A: The method of washers is a technique used to find the volume of a solid formed by revolving a region about an axis. It involves finding the volume of the solid formed by revolving the region about the axis, minus the volume of the solid formed by revolving the region about the axis, plus the volume of the solid formed by revolving the region about the axis.

Q: How do I find the volume of the solid formed by revolving the region about the $y$-axis?

A: To find the volume of the solid formed by revolving the region about the $y$-axis, you will need to use the formula:

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

where $R$ and $r$ are the outer and inner radii of the washer, respectively, and $a$ and $b$ are the limits of integration.

Q: What are the outer and inner radii of the washer?

A: In this case, the outer radius of the washer is given by $R = x^3$, and the inner radius of the washer is given by $r = x^2$.

Q: What are the limits of integration?

A: The limits of integration are $a = 0$ and $b = 1$.

Q: How do I evaluate the integral?

A: To evaluate the integral, you will need to use the power rule of integration, which states that:

$$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$

Q: What is the final answer to the problem?

A: The final answer to the problem is:

$$V = \frac{-2\pi}{35}$$

Rounding to the nearest thousandth, we get:

$$V = -0.179$$

Q: What is the significance of revolving a region about the $y$-axis?

A: Revolving a region about the $y$-axis is a technique used to find the volume of a solid formed by revolving a region about an axis. It has many practical applications in fields such as engineering, physics, and mathematics.

Q: Can I use the method of washers to find the volume of a solid formed by revolving a region about any axis?

A: Yes, you can use the method of washers to find the volume of a solid formed by revolving a region about any axis. However, you will need to adjust the formula and the limits of integration accordingly.

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 outer and inner radii of the washer
• Failing to identify the limits of integration
• Failing to evaluate the integral
• Failing to round the answer to the nearest thousandth

Conclusion

In this article, we have answered some of the most frequently asked questions about revolving a region about the $y$-axis and finding the volume of the resulting solid. We have provided step-by-step instructions on how to use the method of washers to find the volume of a solid formed by revolving a region about the $y$-axis. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of this topic.