Find The Volume Of The Solid Obtained By Rotating The Region Bounded By The Equation $x = -y^2 + 4y - 3$ And $x = 0$ About The Y-axis.

Feb 26, 2025 by ADMIN 135 views

**Volume of a Solid Obtained by Rotating a Region about the Y-axis**

Introduction

In mathematics, the volume of a solid obtained by rotating a region about an axis is a fundamental concept in calculus. The method of disks, also known as the disk method, is used to find the volume of a solid formed by rotating a region about an axis. In this article, we will discuss how to find the volume of a solid obtained by rotating the region bounded by the equation $x = -y^2 + 4y - 3$ and $x = 0$ about the y-axis.

Understanding the Problem

To find the volume of the solid, we need to understand the region bounded by the equation $x = -y^2 + 4y - 3$ and $x = 0$ . The equation $x = -y^2 + 4y - 3$ represents a parabola that opens to the left, and the line $x = 0$ represents the y-axis. The region bounded by these two curves is a parabolic region that extends from the y-axis to the point where the parabola intersects the line $x = 0$ .

Finding the Intersection Points

To find the intersection points of the parabola and the line, we need to set the two equations equal to each other and solve for y. Setting $-y^2 + 4y - 3 = 0$ , we can solve for y using the quadratic formula:

y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, $a = -1$ , $b = 4$ , and $c = -3$ . Plugging these values into the quadratic formula, we get:

y = \frac{-4 \pm \sqrt{4^2 - 4(-1)(-3)}}{2(-1)}

Simplifying the expression, we get:

y = \frac{-4 \pm \sqrt{16 - 12}}{-2}

y = \frac{-4 \pm \sqrt{4}}{-2}

y = \frac{-4 \pm 2}{-2}

Solving for y, we get two possible values:

y = \frac{-4 + 2}{-2} = -1

y = \frac{-4 - 2}{-2} = 3

Finding the Volume of the Solid

To find the volume of the solid, we need to use the method of disks. The method of disks involves integrating the area of the disks formed by rotating the region about the y-axis. 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 distance from the y-axis to the parabola, which is equal to the x-coordinate of the parabola.

Setting Up the Integral

To set up the integral, we need to find the x-coordinate of the parabola as a function of y. The x-coordinate of the parabola is given by the equation:

x = -y^2 + 4y - 3

We can rewrite this equation as:

x = -(y^2 - 4y + 3)

Expanding the expression, we get:

x = -y^2 + 4y - 3

Finding the Volume

To find the volume of the solid, we need to integrate the area of the disks formed by rotating the region about the y-axis. 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 distance from the y-axis to the parabola, which is equal to the x-coordinate of the parabola.

Evaluating the Integral

To evaluate the integral, we need to integrate the area of the disks formed by rotating the region about the y-axis. 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 distance from the y-axis to the parabola, which is equal to the x-coordinate of the parabola.

Simplifying the Expression

To simplify the expression, we need to evaluate the integral:

V = \pi \int_{-1}^{3} (-y^2 + 4y - 3)^2 dy

Expanding the expression, we get:

V = \pi \int_{-1}^{3} (y^4 - 8y^3 + 24y^2 - 36y + 9) dy

Evaluating the Integral

To evaluate the integral, we need to integrate each term separately:

V = \pi \left[ \frac{y^5}{5} - 2y^4 + 8y^3 - 12y^2 + 9y \right]_{-1}^{3}