Problem 75 - Calculus Volume 1

Question

For the following exercises, draw the region bounded by the curves. Then, use the disk method to find the volume when the region is rotated around the \(x\) -axis. \(y=2 x^{2}, x=0, x=4,\) and \(y=0\)

Step-by-Step Solution

Answer

The volume is \(819.2\pi\) cubic units.

Step 1: Identify the Bounded Region

First, outline the region bounded by the given curves. The curves given are: 1. The parabola described by the equation \(y = 2x^2\).2. The vertical line \(x = 0\), which is the y-axis.3. The vertical line \(x = 4\), which is a straight vertical line crossing the x-axis at \(x=4\).4. The horizontal line \(y = 0\), which is the x-axis.The bounded region is between \(x = 0\) and \(x = 4\), above the x-axis, and beneath the parabola \(y = 2x^2\).

Step 2: Setup the Disk Method Formula

The disk method formula for finding the volume \(V\) when rotating a region around the x-axis is given by: \[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]For this problem, the function \(f(x)\) that gives the radius of each disk is \(f(x) = 2x^2\). We rotate this area from \(x = 0\) to \(x = 4\).

Step 3: Set Up the Integral

Substitute \(f(x)\) into the disk method formula:\[ V = \pi \int_{0}^{4} (2x^2)^2 \, dx \]Simplify the integrand: \[(2x^2)^2 = 4x^4\]Thus, the integral becomes:\[ V = \pi \int_{0}^{4} 4x^4 \, dx \]

Step 4: Integrate

Calculate the integral: \[ \pi \int_{0}^{4} 4x^4 \, dx = \pi \left[ \frac{4}{5}x^5 \right]_{0}^{4} \]Evaluate this expression:\[ \pi \left[ \frac{4}{5}(4)^5 - \frac{4}{5}(0)^5 \right] = \pi \left[ \frac{4}{5} \times 1024 \right] = \pi \times 819.2 \]

Step 5: Compute the Volume

Calculate the final volume:\[ V = \pi \times 819.2 = 819.2\pi \]Thus, the volume of the solid generated by rotating the region around the x-axis is \(819.2\pi\) cubic units.

Key Concepts

Volume of Solids of Revolution

The concept of the volume of solids of revolution involves creating a three-dimensional shape by rotating a two-dimensional area around a specified axis. This method helps in finding the volume of the resulting solid. In the given exercise, you are asked to revolve the region bounded by the curves around the x-axis.
To tackle this, the disk method is often used due to its straightforward approach. Consider the shape generated as a series of thin disks stacked upon each other along the axis of rotation, much like stacking slices of bread. The formula utilized to find the volume using the disk method is:\[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]Here, \(V\) symbolizes the volume, and \([f(x)]^2\) represents the area of a cross-sectional disk at any point \(x\). The limits \(a\) and \(b\) are the points along the axis spanning the bounded region. By integrating from \(a\) to \(b\), you accumulate the volume contribution from each disk, resulting in the total volume of the solid of revolution.

Integration Techniques

Integration is a core component of solving problems involving solids of revolution. By applying integration techniques, you determine the area under a curve or, in the case of the disk method, the volume of a solid by evaluating the integral of the area of disks along the axis.
For the given problem, the task is to find the volume when rotating the function \(y = 2x^2\) around the x-axis. This requires:

• First, substituting \(f(x) = 2x^2\) into the disk method formula: \( V = \pi \int_{0}^{4} (2x^2)^2 \, dx \).
• Next, simplifying the expression: \((2x^2)^2 = 4x^4\).
• Hence, the integral becomes: \( V = \pi \int_{0}^{4} 4x^4 \, dx \).
• Performing the integration: \( \pi \left[ \frac{4}{5}x^5 \right]_{0}^{4} \).
• Finally, evaluating the expression at the given bounds: \(\pi \times 819.2\).

The goal is to compute this integral successfully, resulting in the final volume \(819.2\pi\), which reflects the integration techniques applied.

Bounded Regions

Determining the bounded region is critical before applying any method for volumes of revolution. In this exercise, understanding the enclosed area helps properly set up your integral for the volume calculation.
Start by identifying the curves given:

• \(y = 2x^2\): This represents a parabola opening upwards.
• \(x = 0\): The y-axis, acting as a boundary to the left.
• \(x = 4\): A vertical line that limits the region on the right.
• \(y = 0\): The x-axis, serving as the bottom boundary.

These curves together form a region above the x-axis between the lines \(x = 0\) and \(x = 4\) and under the parabola. Recognizing this enclosed shape allows for correct application of the integral setup, ensuring that the disks calculated will wholly contain the volume of revolution. Accurately identifying this bounded region is essential to solving the problem without errors, acting as a fundamental step to ensure precision in subsequent calculations.