Problem 148 - Calculus Volume 1

Question

For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. \(x=y^{2}\) and \(y=x\) rotated around the line \(y=2\)

Step-by-Step Solution

Answer

The volume is approximately \( \frac{32\pi}{15} \) cubic units.

Step 1: Identify the Region

First, we need to identify the region between the curves. The curves given are \( x = y^2 \) and \( y = x \). These intersect at the points (0, 0) and (1, 1), as can be verified by solving the equations together.

Step 2: Set Up the Integral Using the Washer Method

Since we are rotating around the line \( y = 2 \), we will use the washer method where the radii of the washer depend on the distance of the curves from \( y=2 \). The outer radius, from \( y=2 \) to \( y = x \), is \( R(y) = 2 - y \) and the inner radius, from \( y=2 \) to \( x = y^2 \), is \( r(y) = 2 - y^2 \). The volume \( V \) is given by integrating \( \pi \times (\text{outer radius}^2 - \text{inner radius}^2) \).

Step 3: Set Up the Definite Integral

Set up the definite integral for the volume: \[ V = \pi \int_{0}^{1} [(2 - y)^2 - (2 - y^2)^2] \, dy. \] Expand both squared terms to get the integrand: \( (2 - y)^2 = 4 - 4y + y^2 \) and \( (2 - y^2)^2 = 4 - 4y^2 + y^4 \). The integrand thus simplifies to \( y^4 - y^2 + 4y - 4 \).

Step 4: Integrate

Integrate the function \( y^4 - y^2 + 4y - 4 \) with respect to \( y \) from 0 to 1: \[ V = \pi \int_{0}^{1} [y^4 - y^2 + 4y - 4] \, dy \]. The antiderivative is \( \frac{y^5}{5} - \frac{y^3}{3} + 2y^2 - 4y \). Evaluate this from 0 to 1.

Step 5: Evaluate the Integral

Plug the limits into the antiderivative: \[ V = \pi \left(\left[ \frac{1^5}{5} - \frac{1^3}{3} + 2(1)^2 - 4(1) \right] - \left[ \frac{0^5}{5} - \frac{0^3}{3} + 2(0)^2 - 4(0) \right]\right) \]. Calculating this gives \( \pi \left(\frac{1}{5} - \frac{1}{3} + 2 - 4 \right) = \pi \left(\frac{-32}{15}\right) \).

Step 6: Calculate the Volume

Simplify the expression: \( V = -\pi \left(\frac{32}{15}\right) \). The negative sign indicates the region is below \( y = 2 \), as expected from the setup. Thus, the total volume of the solid is about \( \frac{32\pi}{15} \) cubic units.

Key Concepts

Washer Method

The Washer Method is a technique used in Calculus to find the volume of a solid of revolution. Imagine stacking many washers, or hollow discs, to fill up the space of a solid. This method involves calculating the volume of each individual washer and summing them up.

When you rotate a shape around an axis, such as a line parallel to one of the axes, the region between two curves makes up a hollow disk, or washer. Here are the steps we usually follow for the Washer Method:

• Identify the outer and inner curves.
• Determine the outer radius, which is the distance from the axis of rotation to the outer curve.
• Determine the inner radius, which is the distance from the axis of rotation to the inner curve.
• Set up the integral by subtracting the squared inner radius from the squared outer radius, then multiply by \( \pi \).

This integral helps calculate the volume of the solid. Essentially, by working through the integral, you are adding the volume of all those little washers together.

Definite Integral

A definite integral is a fundamental concept in calculus used to calculate the "net area" under a curve within certain bounds. In the context of volume problems, it helps us find the accumulated value, or total volume, over an interval.

When using definite integrals to calculate volumes, especially in solids of revolution, the process is:

• Identify the function or functions whose integral you'd need, representing the area you're revolving.
• Choose definite bounds, or limits of integration, based on where the curves intersect.
• Calculate the integral over these bounds to find the accumulated volume.

In problems involving solids of revolution, you often encounter the form \( \pi \int [R(y)^2 - r(y)^2] \, dy \), where you're essentially finding the volume of one washer at a time and then summing these volumes over the given range.

Volume of Solids of Revolution

The volume of solids of revolution refers to the volume of 3D shapes formed by rotating 2D regions around an axis. This is a core application of integration in calculus and can be visualized as spinning a flat region around an axis and calculating the shape created.

These volumes can be computed using the washer method (or other methods like the disk or shell method) by setting up an integral that revolves around the chosen axis. The steps to calculate these volumes include:

• Identify the curves involved and the axis of rotation.
• Choose the appropriate method—washer, disk, or shell—depending on the geometry of the problem.
• Set up the integral to reflect the area being revolved and compute this over the given interval.

The result of this integration process is the total volume of the solid. Understanding the properties of these revolved solids helps solve many real-world engineering and physics problems, from designing containers to calculating growth rates in biological sciences.