Find The Volume Of The Solid Generated By Revolving The Region Bounded By The Graphs Of The Equations About The $x$-axis:$y = 2x^2$, $y = 0$, $x = 2$.A. $128 \pi$B. $8 \pi$C. $80 \pi$D.

Feb 26, 2025 by ADMIN 186 views

**Finding the Volume of a Solid Generated by Revolving a Region**

Introduction

In mathematics, the method of disks (or washers) is a technique used to find the volume of a solid generated by revolving a region about an axis. This method is based on the concept of slicing the solid into thin disks, each of which has a volume that can be calculated using the formula for the area of a circle. In this article, we will use the method of disks to find the volume of a solid generated by revolving the region bounded by the graphs of the equations $y = 2x^2$ , $y = 0$ , and $x = 2$ about the $x$ -axis.

Understanding the Problem

To begin, let's understand the problem at hand. We are given three equations: $y = 2x^2$ , $y = 0$ , and $x = 2$ . The graph of the equation $y = 2x^2$ is a parabola that opens upwards, while the graph of $y = 0$ is the $x$ -axis. The graph of $x = 2$ is a vertical line that intersects the $x$ -axis at the point $(2, 0)$ . We are asked to find the volume of the solid generated by revolving the region bounded by these three graphs about the $x$ -axis.

Visualizing the Region

To visualize the region bounded by the graphs of the equations, let's first graph the equations on a coordinate plane. The graph of $y = 2x^2$ is a parabola that opens upwards, while the graph of $y = 0$ is the $x$ -axis. The graph of $x = 2$ is a vertical line that intersects the $x$ -axis at the point $(2, 0)$ . The region bounded by these three graphs is a triangular region with vertices at $(0, 0)$ , $(2, 0)$ , and $(2, 4)$ .

Using the Method of Disks

To find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the $x$ -axis, we will use the method of disks. This method involves slicing the solid into thin disks, each of which has a volume that can be calculated using the formula for the area of a circle. The formula for the area of a circle is $A = \pi r^2$ , where $r$ is the radius of the circle.

Calculating the Volume

To calculate the volume of the solid, we will integrate the area of each disk with respect to $x$ . The area of each disk is given by the formula $A = \pi r^2$ , where $r$ is the radius of the disk. In this case, the radius of each disk is given by the equation $y = 2x^2$ . Therefore, the area of each disk is given by the formula $A = \pi (2x^2)^2 = 4 \pi x^4$ .

Evaluating the Integral

To find the volume of the solid, we will integrate the area of each disk with respect to $x$ . The integral is given by the formula $\int_0^2 4 \pi x^4 dx$ . Evaluating this integral, we get:

\int_0^2 4 \pi x^4 dx = \left[ \frac{4 \pi x^5}{5} \right]_0^2 = \frac{4 \pi (2)^5}{5} - \frac{4 \pi (0)^5}{5} = \frac{4 \pi (32)}{5} = \frac{128 \pi}{5}

Simplifying the Answer

To simplify the answer, we can multiply the numerator and denominator by $5$ to get:

\frac{128 \pi}{5} = \frac{128 \pi \cdot 5}{5 \cdot 5} = \frac{640 \pi}{25}

However, this is not one of the answer choices. We can simplify the answer further by multiplying the numerator and denominator by $5$ again to get:

\frac{640 \pi}{25} = \frac{640 \pi \cdot 5}{25 \cdot 5} = \frac{3200 \pi}{125}