#Volume with Disc Method Revolving Around the x-axis

#Table of Contents

1. Introduction to Volume of Revolution
2. Using the Disc Method
3. Exam Tips
4. Worked Example
5. Practice Questions
6. Summary and Key Takeaways
7. Glossary

#Introduction to Volume of Revolution

#What is a Volume of Revolution Around the x-axis?

A solid of revolution is formed when an area bounded by a function $y=f(x)$ (and other boundary equations) is rotated $2\pi$ radians ($360°$) around the $x$-axis. The volume of revolution is the volume of this solid.

Be careful – the 'front' and 'back' of this solid are flat because they were created from straight (vertical) lines. 3D sketches can be misleading.

#Using the Disc Method

#How to Calculate a Volume of Revolution Around the x-axis?

For a continuous function $f$, if the region bounded by:

• the curve $y=f(x)$ and the $x$-axis
• between $x=a$ and $x=b$

is rotated $2\pi$ radians ($360°$) around the $x$-axis, then the volume of revolution is:

$$V = \pi \int_{a}^{b} y^2 , dx = \pi \int_{a}^{b} [f(x)]^2 , dx$$

This method of finding volumes of revolution uses the concept of a definite integral to calculate an accumulation of change. It is a special case of 'finding volumes from areas of known cross-sections'.

For a disc with a circular cross-section of radius $\left|y\right|$ and thickness $dx$, its volume is $\pi y^2 \, dx$. The integral $\pi \int\_{a}^{b} y^2 \, dx$ sums up these infinitesimally thin discs from $x=a$ to $x=b$.

#Exam Tips

#Worked Example

#Example Problem

Find the volume of the solid formed by rotating the region bounded by $y=\sqrt{3x^2+2}$, the $x$-axis, and the vertical line $x=3$ around the $x$-axis. Give your answer as an exact value.

Solution:

Using the formula for the volume of revolution:

$$V = \pi \int_{0}^{3} [\sqrt{3x^2 + 2}]^2 , dx$$

Simplify the integrand:

$$V = \pi \int_{0}^{3} (3x^2 + 2) , dx$$

Integrate:

$$V = \pi \left[ x^3 + 2x \right]_{0}^{3}$$

Evaluate the definite integral:

$$V = \pi \left[ (3^3 + 2 \cdot 3) - (0^3 + 2 \cdot 0) \right]$$

$$V = \pi [27 + 6]$$

$$V = 33\pi$$

Therefore, the volume of the solid is $33\pi , \text{units}^3$.

#Practice Questions

Practice Question

Question 1: Find the volume of the solid formed by rotating the region bounded by $y=x^2$, the $x$-axis, and the lines $x=0$ and $x=2$ around the $x$-axis.

Answer:

$$V = \frac{8\pi}{5} \, \text{units}^3$$

Practice Question

Question 2: Determine the volume of the solid formed by rotating the region bounded by $y=\sin(x)$, the $x$-axis, and the lines $x=0$ and $x=\pi$ around the $x$-axis.

Answer:

$$V = 2\pi^2 \, \text{units}^3$$

#Summary and Key Takeaways

#Summary

• A volume of revolution around the $x$-axis is formed by rotating an area under a curve around the $x$-axis.
• The disc method uses the formula $V = \pi \int_{a}^{b} [f(x)]^2 , dx$ to find this volume.
• Integrating the square of the function from $x=a$ to $x=b$ gives the total volume.

#Key Takeaways

• Always sketch the region if a diagram is not provided.
• Remember that square roots in the function will be squared away in the formula.
• Use the integral limits $x=a$ and $x=b$ as boundaries for your calculations.

#Glossary

Volume of Revolution: The volume of a solid formed by rotating a region around an axis.

Disc Method: A method to calculate the volume of a solid of revolution using integration.

Integral: A mathematical concept that represents the area under the curve, accumulation of quantities, etc.

Definite Integral: An integral with specific upper and lower limits, used to calculate exact areas or volumes.

#Practice Questions

Practice Question

Question 3: Find the volume of the solid formed by rotating the region bounded by $y=2x$, the $x$-axis, and the lines $x=0$ and $x=1$ around the $x$-axis.

Answer:

$$V = \frac{2\pi}{3} \, \text{units}^3$$