Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the volume of the solid generated by revolving about the \(y\)-axis the region bounded by the line \(y=4 x\) and the parabola \(y=4 x^{2}\).

Step-by-Step Solution

Answer

The volume of the solid is \(\frac{2\pi}{3}\).

Step 1: Identify the Intersection Points

We need to find the points where the line and the parabola intersect. Set the equations equal to each other: \(4x = 4x^2\). Simplifying this gives \(x(x - 1) = 0\), so the intersection points are at \(x = 0\) and \(x = 1\).

Step 2: Set Up the Integral for the Volume

Since we are revolving around the y-axis, we will use the method of cylindrical shells. The volume \(V\) of the solid can be calculated using the formula: \[ V = 2\pi \int_{a}^{b} x \, (f(x) - g(x)) \, dx \]where \( f(x) = 4x \) and \( g(x) = 4x^2 \), \(a = 0\), and \(b = 1\).

Step 3: Substitute Functions into the Integral

Substitute \(f(x) = 4x\) and \(g(x) = 4x^2\) into the volume integral: \[ V = 2\pi \int_{0}^{1} x \, (4x - 4x^2) \, dx \]Simplify the integrand: \[ V = 2\pi \int_{0}^{1} x \, (4x - 4x^2) \, dx = 2\pi \int_{0}^{1} (4x^2 - 4x^3) \, dx \]

Step 4: Integrate the Function

Calculate the integral: \[ 2\pi \int_{0}^{1} (4x^2 - 4x^3) \, dx = 2\pi \left[ \frac{4x^3}{3} - \frac{4x^4}{4}\right]_{0}^{1} \]Evaluate this from \(0\) to \(1\):\[ = 2\pi \left( \frac{4(1)^3}{3} - \frac{4(1)^4}{4} \right) \]\[ = 2\pi \left( \frac{4}{3} - 1 \right) \]

Step 5: Calculate the Result

Simplify the expression:\[ = 2\pi \left( \frac{4}{3} - 1 \right) = 2\pi \left( \frac{4}{3} - \frac{3}{3} \right) = 2\pi \left( \frac{1}{3} \right) \]\[ = \frac{2\pi}{3} \]

Key Concepts

Volume of Solids of Revolution

When you revolve a 2D region around an axis, such as the y-axis, you create a 3D object called a solid of revolution. This concept is common in calculus, as it involves calculating the volume of these solids. The basic idea is to find how much space the solid occupies when the specified region is spun around the axis.
One technique to find this volume is known as the cylindrical shells method. It is particularly useful when the solid is revolving around an axis that is not in its principal orientation, like the y-axis for certain equations.
Understanding the volume of solids of revolution helps in various real-world applications:

• Engineering projects, where the measurement of symmetrical structures is needed.
• Designing objects, such as vessels or containers that have circular shapes.

Cylindrical Shells Method

The cylindrical shells method is a technique in integral calculus used to find the volume of a solid of revolution. Unlike other methods, this approach is beneficial when revolving a region around a vertical line, like the y-axis.
Here's how it works:

• A thin 'shell' is considered, much like a hollow cylinder, within the solid.
• The shell has a height and a radius corresponding to the distance from the axis of rotation.
• The shell's thickness is an infinitesimally small width, represented as dx in the integral.
• The volume of each shell is calculated, and then the integral sums up all these infinitesimal contributions.

For the exercise given above, the formula \( V = 2\pi \int_{a}^{b} x (f(x) - g(x)) \, dx \) is used. Here, \(a\) and \(b\) mark the interval of integration, \(f(x)\) and \(g(x)\) are the function values that define the heights of the shells.
By using cylindrical shells, you can effectively handle cases where revolutions and curves may seem complex when tackled with the disk or washer methods.

Integral Calculus

Integral calculus, a major part of calculus, is all about summation and finding the total values like areas, volumes, etc., over a continuous domain. It's opposite to differential calculus, which focuses on instantaneous rates of change.
In practice, integral calculus is all about accumulation.

• It is used to compute areas under curves, which is essential for determining quantities like total distance, area, or even the total income over a specific period.
• It applies to calculating volumes, like in the exercise provided, where the volume of a solid created by rotating a region around an axis is found.

The definite integral, which has upper and lower limits, describes the accumulation process over a specific segment and is represented mathematically by \( \int_{a}^{b} f(x) \, dx \).
In this integration, each small slice or segment within the total area or volume is considered and added up to provide a complete solution.

Intersection Points of Curves

To find the volume of the solid of revolution, one must first identify the intersection points of the curves that define the boundaries. These intersection points give us the limits for calculating the volume.
In the problem, you set the equations equal to solve where the line \(y=4x\) intersects the parabola \(y=4x^2\). Solving the equation \(4x = 4x^2\) simplifies to finding the roots of \(x(x - 1) = 0\), which results in intersection at \(x = 0\) and \(x = 1\).
These values are crucial:

• They serve as the integration bounds \(a\) and \(b\) to compute volumes.
• They define the range of the solid being rotated to obtain the resultant 3D shape.

Understanding these points not only helps pinpoint where curves meet but also sets the stage for calculating precise areas and volumes in calculus.