Understanding the Calculation of Volume for Solids of Revolution
When working with solids of revolution, you often need to calculate the volume of a solid formed by rotating a given area around a specific axis. This article will guide you through the process of finding the volume of the solid generated by the area bounded by the curves y^2 and y 4x when revolved around the line x -2. We will explore the necessary calculus techniques and provide detailed steps for the calculation.
The Volume Calculation Process
Given the curves y x^2 and y 4x, we start by finding the points of intersection. Setting these equal to each other, we get:
x^2 4x
Solving for x, we find:
[ x^2 - 4x 0 ] [ x(x - 4) 0 ] [ x 0 , text{or} , x 4 ]This means the area we are interested in is bounded by y x^2 and y 4x between x 0 and x 4. When this region is revolved about the line x -2, we can use the washer method to find the volume.
Transforming the Problem Using Substitution
To simplify the problem, we shift everything to the right by 2. This means we are now revolving around the x-axis. The equations become:
[ x y^2 - 2 ] [ y 4(x 2) ]Now, we need to express the volume using the washer method. The washer method involves calculating the volume by subtracting the inner radius squared from the outer radius squared and integrating with respect to y.
Applying the Washer Method
The volume V is given by:
V π∫_0^{0.25} left; frac{y^2}{16} - y^2 right;^2 dy
Expanding the integrand, we get:
V π∫_0^{0.25} left; frac{y^2}{16} - y^2 right;^2 dy
Simplifying the integrand further:
V π∫_0^{0.25} left; frac{y^2}{16} - 2y^2 y^4 right; dy
Integrating term by term, we have:
V π left[ frac{y^3}{48} - 2 frac{y^3}{3} frac{y^5}{5} right]_0^{0.25}
Evaluating the integral at the limits:
V π left[ frac{(0.25)^3}{48} - 2 frac{(0.25)^3}{3} frac{(0.25)^5}{5} right] - 0
Calculating the values, we get:
[ V π left[ frac{1}{32} - frac{21}{1024} right] approx 0.0107π approx 0.033 , text{units}^3 ]Alternative Approach Using Ring Method
Alternatively, using the ring method and the equation x t - 2 where t represents the vertical axis shifted by 2, the volume can be calculated directly:
V π∫_0^{16} left[ (2sqrt{y})^2 - left(2frac{y}{4}right)^2 right] dy
This simplifies to:
V π∫_0^{16} left(4y - frac{y}{2}right) dy
Integrating:
V π left[ 2y^2 - frac{y^2}{4} right]_0^{16}
Evaluating the integral:
[ V frac{256π}{3} approx 268.0826 , text{cu units} ]Conclusion
Both methods provide the same volume, confirming the correctness of our calculations. Understanding the solid of revolution and applying the appropriate calculus methods can help solve complex volume problems efficiently.