Question

Find the volume of the solid generated by revolving about the line \(y=2\) the region in the first quadrant bounded by the parabolas \(3 x^{2}-16 y+48=0\) and \(x^{2}-16 y+80=0\) and the \(y\) -axis.

Short answer

The volume of the solid is \(16\pi\).

Step-by-step solution

Simplify the equations of the parabolas

First, simplify the equations of the parabolas by expressing them in terms of the standard parabola equation. For the first parabola: rearrange to get \( y = \frac{3}{16}x^2 + 3 \). For the second parabola, rearrange to get \( y = \frac{1}{16}x^2 + 5 \).

Identify the region of integration

The region of interest is where both parabolas meet in the first quadrant. Set \( \frac{3}{16}x^2 + 3 = \frac{1}{16}x^2 + 5 \) to find the points of intersection. Solving, we get \( x=4 \) and \( x=0 \) at \( y = 5 \). The region is bounded between \( x=0 \) and \( x=4 \).

Sketch the region and determine radii of washers

When rotated about the line \( y = 2 \), the region forms washers. The outer radius \( R(y) = 2 + y - 3 \) is given by \( R(y) = y - 1 \) derived from the first parabola. The inner radius \( r(y) = 2 + y - 5 \) is given by \( r(y) = y - 3 \) derived from the second parabola.

Set up the integral for the volume using the washer method

The volume \( V \) of the solid formed is given by the integral of the area of the washer cross-sections: \[ V = \pi \int_{y=3}^{5} ((y - 1)^2 - (y - 3)^2) \, dy \]

Compute the integral

Simplify the integrand: \((y - 1)^2 - (y - 3)^2 = (y^2 - 2y + 1) - (y^2 - 6y + 9) = 4y - 8\.\) The integral becomes: \[ V = \pi \int_{3}^{5} (4y - 8) \, dy \]Calculate the integral:\[V = \pi \left[ 2y^2 - 8y \right]_{3}^{5} \]\[= \pi \left( [2(5)^2 - 8(5)] - [2(3)^2 - 8(3)] \right) \]\[= \pi (2 \cdot 25 - 40 - (2 \cdot 9 - 24)) \]\[= \pi (50 - 40 - 18 + 24) \]\[= 16\pi \].

Conclusion

Thus, the volume of the solid is \( 16\pi \).

Key concepts

Volume of Revolution

When dealing with calculus, one fascinating concept is finding the volume of a solid that is created by revolving a region around a line. This is often called the "volume of revolution." Imagine spinning a flat shape around a line, similar to how a potter spins clay on a wheel. The resulting shape can be visualized like a pot or vase. Calculating this volume involves integrating over the area of cross-sections of the shape. In our exercise, the solid is generated by rotating the region between two parabolas around the line \( y = 2 \).
This method helps find the total space the solid occupies, which is a crucial aspect in many engineering and physics applications where spatial dimensions are critical. By understanding how this works, you can solve real-world problems like fluid dynamics or structural analysis.

Parabola Equations

Parabolas are a type of curve, and they have a specific mathematical representation. The general form of a parabola equation opens either upwards or downwards, described by \( y = ax^2 + bx + c \). In this exercise, we simplify and manipulate the given parabola equations to understand their orientation and position in the coordinate plane.
The two parabolas given are \(3x^2 - 16y + 48 = 0\) and \(x^2 - 16y + 80 = 0\). By rearranging, we get the standard forms:

• \( y = \frac{3}{16}x^2 + 3 \)
• \( y = \frac{1}{16}x^2 + 5 \)

These forms help us identify how the parabolas interact with the y-axis and each other. Understanding parabola equations is essential as it lays the foundation for determining the regions and limits of integration. By knowing where these curves intersect, we establish the boundaries necessary for calculating areas and volumes.

Washer Method

The Washer Method is a precise tool for calculating the volume of solids of revolution, especially when there are hollow sections. When rotating shapes around an axis, the resulting solid often looks like a stack of washers. Each washer has an outer radius and an inner radius, creating a donut-like cross-section.
In our problem, the region is revolved around the line \( y = 2 \). The washers' radii need to be defined based on their distance from this line. The outer radius is given by \( R(y) = y - 1 \), while the inner radius is \( r(y) = y - 3 \).
The difference between the squares of these radii \((R^2 - r^2)\) gives the area of the washer, and integrating this over the specified interval will give us the entire volume. This approach is highly effective whenever you're asked to find a volume created by revolving areas that do not touch the axis of revolution directly.

Integral Calculation

Integral calculation is a fundamental process in calculus that enables us to find areas, volumes, and more. In the context of the volume of revolution, we use integral calculus to sum up the infinite number of tiny areas to get the final volume.
For this exercise, we set up the integral based on the washer method:
\[ V = \pi \int_{3}^{5} ((y - 1)^2 - (y - 3)^2) \, dy \]
Simplifying this expression properly is crucial. The integrand \((4y - 8)\) is obtained after simplification.
Calculating the definite integral gives:

• Evaluate the antiderivative: \(2y^2 - 8y\)
• Calculate from \(y = 3\) to \(y = 5\).

Ultimately, the evaluated integral \([2(5)^2 - 8(5)] - [2(3)^2 - 8(3)]\) results in a final volume of \(16\pi\). This step is vital for arriving at the correct conclusion and ensures the volume derived accurately reflects the physical solid.