Finding the Volume by Slicing

Oct 13, 2025 by talynt.org 89 views

Introduction

In this article, we will explore the concept of finding the volume of a solid obtained by rotating a region about the y-axis. The region is bounded by two curves, $y = \sqrt{16x}$ and $y = x^2$ . We will use the method of slicing to find the volume of the solid.

Understanding the Problem

To find the volume of the solid, we need to understand the region that is being rotated. The region is bounded by two curves, $y = \sqrt{16x}$ and $y = x^2$ . We can see that the curve $y = \sqrt{16x}$ is a parabola that opens upwards, while the curve $y = x^2$ is also a parabola that opens upwards.

Finding the Volume by Slicing

To find the volume of the solid, we will use the method of slicing. This method involves dividing the region into thin slices, and then finding the volume of each slice. We will then sum up the volumes of all the slices to find the total volume of the solid.

Step 1: Define the Limits of Integration

The first step in finding the volume of the solid is to define the limits of integration. The region is bounded by the curves $y = \sqrt{16x}$ and $y = x^2$ . We can see that the curve $y = \sqrt{16x}$ intersects the curve $y = x^2$ at the point $(4, 4)$ . Therefore, the limits of integration are from $x = 0$ to $x = 4$ .

Step 2: Find the Volume of Each Slice

The next step is to find the volume of each slice. We can do this by finding the area of each slice, and then multiplying it by the thickness of the slice. The area of each slice is given by the difference between the two curves, $y = \sqrt{16x}$ and $y = x^2$ . We can find this difference by subtracting the two curves.

Step 3: Integrate the Volume of Each Slice

Once we have found the volume of each slice, we need to integrate it to find the total volume of the solid. We can do this by summing up the volumes of all the slices.

Step 4: Evaluate the Integral

The final step is to evaluate the integral to find the total volume of the solid. We can do this by using the fundamental theorem of calculus.

Calculating the Volume

Now that we have defined the limits of integration, found the volume of each slice, and integrated the volume of each slice, we can calculate the volume of the solid.

The volume of the solid is given by the integral:

V = \int_{0}^{4} \pi \left( \sqrt{16x} \right)^2 - \left( x^2 \right)^2 dx

Evaluating the integral, we get:

V = \int_{0}^{4} \pi \left( 16x \right) - \left( x^4 \right) dx

V = \pi \left[ 8x^2 - \frac{x^5}{5} \right]_{0}^{4}

V = \pi \left[ 8(4)^2 - \frac{(4)^5}{5} \right]

V = \pi \left[ 128 - \frac{1024}{5} \right]

V = \pi \left[ 128 - 204.8 \right]

V = \pi \left[ -76.8 \right]

V = -76.8 \pi

However, since the volume cannot be negative, we take the absolute value of the result:

V = 76.8 \pi

Conclusion

In this article, we have found the volume of the solid obtained by rotating about the y-axis the region bounded by $y = \sqrt{16x}$ and $y = x^2$ . We used the method of slicing to find the volume of the solid. The volume of the solid is given by $V = 76.8 \pi$ .

Discussion

The method of slicing is a powerful tool for finding the volume of solids. It involves dividing the region into thin slices, and then finding the volume of each slice. We can then sum up the volumes of all the slices to find the total volume of the solid. In this article, we have used the method of slicing to find the volume of the solid obtained by rotating about the y-axis the region bounded by $y = \sqrt{16x}$ and $y = x^2$ .

References

[1] Calculus, 3rd edition, Michael Spivak
[2] Calculus, 2nd edition, James Stewart
[3] Calculus, 1st edition, Michael Spivak

Diagram

The diagram below shows the region bounded by $y = \sqrt{16x}$ and $y = x^2$ .

Diagram

Note: The diagram is not to scale.

Introduction

In our previous article, we explored the concept of finding the volume of a solid obtained by rotating a region about the y-axis. We used the method of slicing to find the volume of the solid bounded by $y = \sqrt{16x}$ and $y = x^2$ . In this article, we will answer some frequently asked questions about finding the volume of a solid obtained by rotating a region about the y-axis.

Q1: What is the method of slicing?

A1: The method of slicing is a technique used to find the volume of a solid obtained by rotating a region about the y-axis. It involves dividing the region into thin slices, and then finding the volume of each slice. We can then sum up the volumes of all the slices to find the total volume of the solid.

Q2: How do I find the limits of integration?

A2: To find the limits of integration, we need to determine the points where the two curves intersect. We can do this by setting the two curves equal to each other and solving for x.

Q3: What is the formula for finding the volume of a solid obtained by rotating a region about the y-axis?

A3: The formula for finding the volume of a solid obtained by rotating a region about the y-axis is given by:

V = \int_{a}^{b} \pi \left( f(x) \right)^2 - \left( g(x) \right)^2 dx

where $f(x)$ and $g(x)$ are the two curves that bound the region.

Q4: How do I evaluate the integral?

A4: To evaluate the integral, we need to use the fundamental theorem of calculus. We can do this by using the power rule of integration and the constant multiple rule of integration.

Q5: What is the significance of the absolute value in the formula for finding the volume of a solid obtained by rotating a region about the y-axis?

A5: The absolute value in the formula is used to ensure that the volume of the solid is always positive. This is because the volume of a solid cannot be negative.

Q6: Can I use the method of slicing to find the volume of a solid obtained by rotating a region about the x-axis?

A6: Yes, you can use the method of slicing to find the volume of a solid obtained by rotating a region about the x-axis. However, you will need to use a different formula, which is given by:

V = \int_{a}^{b} \pi \left( f(y) \right)^2 - \left( g(y) \right)^2 dy

Q7: What are some common mistakes to avoid when finding the volume of a solid obtained by rotating a region about the y-axis?

A7: Some common mistakes to avoid when finding the volume of a solid obtained by rotating a region about the y-axis include:

Failing to determine the limits of integration correctly
Failing to evaluate the integral correctly
Failing to use the absolute value in the formula
Failing to use the correct formula for finding the volume of a solid obtained by rotating a region about the y-axis

Q8: Can I use a calculator to find the volume of a solid obtained by rotating a region about the y-axis?

A8: Yes, you can use a calculator to find the volume of a solid obtained by rotating a region about the y-axis. However, you will need to enter the correct formula and limits of integration into the calculator.

Q9: What are some real-world applications of finding the volume of a solid obtained by rotating a region about the y-axis?

A9: Some real-world applications of finding the volume of a solid obtained by rotating a region about the y-axis include:

Calculating the volume of a tank or a container
Calculating the volume of a solid object
Calculating the volume of a region in three-dimensional space

Q10: Can I use the method of slicing to find the volume of a solid obtained by rotating a region about the y-axis in three-dimensional space?

A10: Yes, you can use the method of slicing to find the volume of a solid obtained by rotating a region about the y-axis in three-dimensional space. However, you will need to use a more complex formula, which is given by:

V = \int_{a}^{b} \pi \left( f(x,y) \right)^2 - \left( g(x,y) \right)^2 dx dy

Conclusion

In this article, we have answered some frequently asked questions about finding the volume of a solid obtained by rotating a region about the y-axis. We have covered topics such as the method of slicing, finding the limits of integration, and evaluating the integral. We have also discussed some common mistakes to avoid and some real-world applications of finding the volume of a solid obtained by rotating a region about the y-axis.