#Volume with Disc Method Revolving Around the y-axis

#Table of Contents

1. Introduction to Volume of Revolution Around the y-axis
2. Disc Method for Calculating Volume
3. Exam Tips
4. Worked Example
5. Practice Questions
6. Glossary
7. Summary and Key Takeaways

#Introduction to Volume of Revolution Around the y-axis

A volume of revolution around the y-axis is similar to a volume of revolution around the x-axis. It involves rotating an area bounded by a function $y = f(x)$ and other boundary equations around the y-axis to form a solid shape. This shape's volume is what we are interested in calculating.

#Disc Method for Calculating Volume

To calculate the volume of revolution around the y-axis using the disc method, follow these steps:

1. Identify the boundaries: Determine the region bounded by the curve and the y-axis between $y = a$ and $y = b$.

2. Rewrite the function: Express the function $y = f(x)$ as $x = g(y)$.

3. Set up the integral: The volume $V$ of the solid of revolution is given by: $$V = \pi \int_{a}^{b} x^2 , dy = \pi \int_{a}^{b} (g(y))^2 , dy$$

4. Evaluate the integral: Perform the integration with respect to $y$.

If $y = a$ and $y = b$ are not explicitly stated in a problem, the boundaries could involve the x-axis ($y = 0$) or another function $y = f(x)$.

#Exam Tips

1. Square Roots: If the function has a square root, it will be squared away in the integration process.
2. Sketching: If no diagram is provided, sketching the curve, limits, etc., can be very helpful. A graphing calculator can assist with this.

#Worked Example

Problem: Calculate the volume of the solid formed by rotating the area bounded by $y = \arcsin(2x+1)$, the x-axis, and the positive y-axis about the y-axis. Give your answer correct to 3 decimal places.

Solution:

1. Rewrite the function: $$y = \arcsin(2x+1)$$ Solve for $x$: $$\sin(y) = 2x + 1 \implies x = \frac{\sin(y) - 1}{2}$$

2. Set up the integral: $$V = \pi \int_{0}^{\frac{\pi}{2}} \left( \frac{\sin(y) - 1}{2} \right)^2 , dy$$

3. Evaluate the integral using a calculator: $$V = \pi \int_{0}^{\frac{\pi}{2}} \left( \frac{\sin(y) - 1}{2} \right)^2 , dy \approx 0.279754...$$ Rounded to three decimal places: $$V \approx 0.280 , \text{units}^3$$

#Practice Questions

Practice Question

1. Calculate the volume of the solid formed by rotating the area bounded by $y = \sqrt{x}$, the y-axis, and the line $y=2$ about the y-axis.

Practice Question

2. Find the volume of the solid obtained by rotating the region bounded by $y = x^2$, the x-axis, and $y=4$ around the y-axis.

#Glossary

• Volume of Revolution: The volume of a solid formed by rotating a region around an axis.
• Disc Method: A method for finding the volume of a solid of revolution by integrating cross-sectional areas perpendicular to the axis.
• Integral: A mathematical concept used to find areas, volumes, and totals.

#Summary and Key Takeaways

• A volume of revolution around the y-axis is calculated by rotating a region bounded by a function and other boundaries around the y-axis.
• The disc method involves setting up an integral with respect to $y$ and rewriting the function $x$ in terms of $y$.
• Always verify the limits of integration and ensure the function is correctly expressed.
• Practice problems help solidify understanding and application of the disc method for volume calculations.

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By following these structured notes, students should find it easier to grasp the concept of volumes of revolution around the y-axis and effectively prepare for their exams.