Step 1: Concept Overview:
The volume of a solid formed by rotating the region between two curves around the x-axis is calculated using the washer method. This method involves integrating the difference between the areas of the outer and inner circles along the axis of revolution.
Step 2: Formula:
The washer method formula for volume \(V\) when revolving around the x-axis is:
\[ V = \pi \int_{a}^{b} [R(x)^2 - r(x)^2] dx \]
Here, \(R(x)\) represents the outer radius, and \(r(x)\) represents the inner radius.
Step 3: Detailed Solution:
First, rewrite the equations in terms of \(x\):\(X = \sqrt{Y} \implies Y = X^2\)\(X = \frac{Y}{4} \implies Y = 4X\)Using \(x\) and \(y\): \(y = x^2\) and \(y = 4x\).
Find the intersection points by equating the functions:\[ x^2 = 4x \]\[ x^2 - 4x = 0 \]\[ x(x - 4) = 0 \]The curves intersect at \(x = 0\) and \(x = 4\), defining the interval \([0, 4]\).
Determine which function is greater (outer radius) within \((0, 4)\). Test \(x=1\):For \(y = 4x\), \(y = 4(1) = 4\).
For \(y = x^2\), \(y = 1^2 = 1\).
Since \(4x>x^2\) in \((0, 4)\), \(R(x) = 4x\) and \(r(x) = x^2\).
Set up and solve the volume integral:\[ V = \pi \int_{0}^{4} [(4x)^2 - (x^2)^2] dx \]\[ V = \pi \int_{0}^{4} (16x^2 - x^4) dx \]\[ V = \pi \left[ \frac{16x^3}{3} - \frac{x^5}{5} \right]_{0}^{4} \]\[ V = \pi \left[ \left(\frac{16(4)^3}{3} - \frac{(4)^5}{5}\right) - \left(\frac{16(0)^3}{3} - \frac{(0)^5}{5}\right) \right] \]\[ V = \pi \left[ \frac{16(64)}{3} - \frac{1024}{5} \right] \]\[ V = \pi \left[ \frac{1024}{3} - \frac{1024}{5} \right] \]\[ V = 1024\pi \left[ \frac{1}{3} - \frac{1}{5} \right] = 1024\pi \left[ \frac{5 - 3}{15} \right] = 1024\pi \left[ \frac{2}{15} \right] \]\[ V = \frac{2048\pi}{15} \]
Step 4: Result:
The volume is \(\frac{2048\pi}{15}\) cubic units.