Problem 17 - Calculus: Early Transcendentals

Question

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. \( x = y^2 \) , \( x = 1 - y^2 \) ; about \( x = 3 \)

Step-by-Step Solution

Answer

The volume is \( \frac{22\pi}{3} \).

Step 1: Sketch the Region

Draw the parabolas defined by the equations \( x = y^2 \) and \( x = 1 - y^2 \) on the coordinate plane. These curves intersect at \( y = -1 \) and \( y = 1 \). The region enclosed by these curves forms a symmetrical lens shape along the \( y \)-axis.

Step 2: Setup the Disk Method

To find the volume of the solid formed by rotating this region around the line \( x = 3 \), we'll use the washer method. Consider a typical vertical slice of thickness \( dy \) at a height \( y \), which forms a washer. The outer radius \( R \) of the washer is \( 3 - (-y^2 + 1) \) and the inner radius \( r \) is \( 3 - y^2 \).

Step 3: Determine Radii of the Washers

The radius of the outer circle is \( R = 3 - (1 - y^2) = 2 + y^2 \) and the radius of the inner circle is \( r = 3 - y^2 \). The area of the washer is given by \[ A(y) = \pi (R^2 - r^2) = \pi ((2 + y^2)^2 - (3 - y^2)^2) \].

Step 4: Calculate the Area of the Washer

Expanding the squares gives: \((2 + y^2)^2 = 4 + 4y^2 + y^4\) and \((3 - y^2)^2 = 9 - 6y^2 + y^4\). Thus, the area function simplifies to \( A(y) = \pi (4 + 4y^2 + y^4 - 9 + 6y^2 - y^4) \).

Step 5: Simplify the Area Expression

Combine like terms to simplify \( A(y) \): \[ A(y) = \pi (4y^2 - 5) \].

Step 6: Integrate to Find Volume

Integrate the area of the washer along the \( y \)-axis from \(-1\) to \( 1 \) to find the volume: \[ V = \int_{-1}^{1} \pi (4y^2 - 5) \, dy \].

Step 7: Evaluate the Integral

Calculate the definite integral: \[ V = \pi \left[ \frac{4}{3}y^3 - 5y \right]_{-1}^{1} \]. Evaluating gives: \[ V = \pi \left( \frac{4}{3}(1)^3 - 5(1) - \left( \frac{4}{3}(-1)^3 - 5(-1) \right) \right) \].

Step 8: Final Computation

Simplifying, we find: \[ V = \pi \left( \frac{4}{3} - 5 - \left( -\frac{4}{3} + 5 \right) \right) = \pi \left( \frac{8}{3} - 10 \right) = \pi \left( \frac{-22}{3} \right) = \frac{-22\pi}{3} \]. Since volume must be positive, we take the absolute value.

Key Concepts

Disk Method

The disk method is a way to calculate the volume of a solid of revolution. This method involves slicing the solid perpendicular to the axis of rotation into thin, disk-like slices. Each slice is essentially a circle, and the basic idea is to sum up the volumes of these slices to get the total volume.When using the disk method, we follow these steps:

• Identify the region that is being rotated.
• Determine the axis of rotation.
• Slice the solid perpendicular to the axis at a typical position.
• Calculate the radius of each slice, which is the distance from the edge of the region to the axis of rotation.
• The volume of each slice is the area of the circular face times the thickness of the slice.

The formula for the volume using the disk method is:\[ V = \int_{a}^{b} \pi [f(y)]^2 \, dy \]where \( f(y) \) is the function representing the radius of the disk.

Washer Method

The washer method is an extension of the disk method. It is used when the solid has a hole in the center, like a doughnut, as opposed to being completely solid. The solid of revolution is sliced into washers instead of disks, each having an outer radius and an inner radius.Here's how you can understand and apply the washer method:

• Define the outer and inner curves that bound the region being rotated.
• Calculate the outer radius \( R \) and the inner radius \( r \) of the washer as the distances from these curves to the axis of rotation.
• The area of a washer is given by the difference between the areas of the outer and inner circles: \( A(y) = \pi (R^2 - r^2) \).
• Integrate this area formula across the bounds of the region to get the volume of the solid.

Using the washer method, the volume is computed with the formula:\[ V = \int_{a}^{b} \pi (R^2 - r^2) \, dy \]This formula accounts for the hollow center by subtracting the volume of the inner cylinder from the volume of the outer cylinder.

Integral Calculus

Integral calculus plays a crucial role in finding the volumes of solids of revolution. It allows us to add up the infinitely small volumes of disks or washers that make up the solid.This branch of calculus is concerned with the concept of integration, which can be thought of as the process of finding the area under a curve. In the context of volume calculation, integration sums up the infinitesimally small elements (e.g., areas of disks or washers) along a continuous range.Key points for understanding integral calculus in solving volume problems include:

• Use definite integrals for calculating the total volume, since they provide the sum from point \( a \) to point \( b \).
• Recognize that the limits of integration correspond to the bounds of the region being rotated.
• Apply the fundamental theorem of calculus to evaluate definite integrals and obtain the final volume.

Integral calculus thus provides a powerful tool for solving geometric problems involving volumes, areas, and lengths, turning complex shapes into manageable computations.