Volume Calculations by Revolving Around the y-Axis: A Comprehensive Guide
Understanding the volume of a solid formed through revolution is a key concept in calculus, particularly with methods such as the washer method. This article delves into a detailed analysis of calculating the volume formed by revolving the area bounded by the curve y^2 4x - y - 1 and the line x 1 about the y-axis.
Revisiting the Given Functions
The given functions are:
y^2 4x - y - 1 x 1To simplify the first function, we rewrite y^2 4x - y - 1 as:
y^2 y - 4x 1 0
This is a quadratic equation in y. Solving for y, we get:
y^2 y - 4x 1 0 rarr; y -frac{1 pm sqrt{1 16x - 4}}{2}
Graphing the Functions and Marking the Region Bounded by Them
The graph of the equation y^2 4x - y - 1 can be visualized with the line x 1. The region we are interested in is the area bounded by these two functions.
Volume Using the Washer Method
The washer method involves finding the volume of each disk with an outer radius and a corresponding disk with an inner radius, then adding them together through integration.
Volume of Each Disk
The volume of each outer disk is given by:
pi r_{text{o}}^2 t
The outer radius r_{text{o}} is the distance from the y-axis to the line x 1, so:
r_{text{o}} 1 - 0 1
The volume of the inner disk is:
pi r_{text{i}}^2 t
The inner radius r_{text{i}} is the distance from the y-axis to the curve y^2 4x - y - 1:
r_{text{i}} frac{y^2}{4} - 0 frac{y^2}{4}
The thickness t is dx.
Limits of Integration
To find the limits of integration, we substitute x 1 into the equation y^2 4x - y - 1:
y^2 4 - y - 1 rarr; y^2 y - 3 0
Solving for y, we get:
y frac{-1 pm sqrt{1 12}}{2} frac{-1 pm sqrt{13}}{2}
The limits of integration are from y - frac{1 sqrt{13}}{2}) to (y frac{-1 - sqrt{13}}{2}), but considering the simpler bounds for our purposes, we use (-4) to 0.
Setting Up the Integration
The volume is given by the integral:
V pi displaystyleint_{-4}^0 left(1 - frac{y^2}{4}right)^2 dy
Expanding the integrand:
V pi displaystyleint_{-4}^0 left(1 - frac{y^2}{2} - frac{y^4}{16}right) dy
Integrating term by term:
V pi left[y - frac{y^3}{6} - frac{y^5}{80}right]_{-4}^0
Evaluating the integral:
V pi left[0 - left(-4 frac{(-4)^3}{6} - frac{(-4)^5}{80}right)right]
Simplifying this:
V pi left[4 - frac{-64}{6} frac{1024}{80}right] pi left[4 frac{32}{3} frac{128}{10}right] pi left[frac{64}{15}right] frac{16pi}{5}
Thus, the volume of the solid formed by revolving the given region about the y-axis is frac{16pi}{5} cubic units. Alternatively, the volume can be computed as:
V frac{4pi}{5}
This calculation and the alternate form using the washer method demonstrate the effective use of integration techniques in volume calculations.