Question

The volume of the solid for the region enclosed by the curves \(X = \sqrt{Y}\), \(X = \frac{Y}{4}\) revolve about x-axis, is

• A. \(\frac{2048\pi}{15}\) cubic units
• B. \(\frac{1024\pi}{15}\) cubic units
• C. \(\frac{4\pi}{15}\) cubic units
• D. \(\frac{512\pi}{15}\) cubic units

Hint:

When using the washer or disk method, always sketch the region or test a point to correctly identify the outer radius \(R(x)\) and inner radius \(r(x)\). A common mistake is swapping them, which results in a negative volume.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To find the volume of a solid generated by revolving a region between two curves about the x-axis, we use the washer method. The volume is found by integrating the difference of the areas of two circles (outer and inner radii) along the axis of revolution.

Step 2: Key Formula or Approach:
The volume \(V\) by the washer method for revolution about the x-axis is given by: \[ V = \pi \int_{a}^{b} [R(x)^2 - r(x)^2] dx \] where \(R(x)\) is the outer radius and \(r(x)\) is the inner radius.

Step 3: Detailed Explanation:
First, we express the curves as functions of \(x\). \(X = \sqrt{Y} \implies Y = X^2\) \(X = \frac{Y}{4} \implies Y = 4X\) Let's use \(x\) and \(y\) for variables: \(y = x^2\) and \(y = 4x\).
Next, we find the points of intersection by setting the functions equal to each other: \[ x^2 = 4x \] \[ x^2 - 4x = 0 \] \[ x(x - 4) = 0 \] The points of intersection are at \(x = 0\) and \(x = 4\). So, our integration interval is \([0, 4]\).
In the interval \((0, 4)\), we determine which function is on top (outer radius). Let's test \(x=1\): For \(y = 4x\), \(y = 4(1) = 4\).
For \(y = x^2\), \(y = 1^2 = 1\).
Since \(4x>x^2\) on \((0, 4)\), the outer radius is \(R(x) = 4x\) and the inner radius is \(r(x) = x^2\).
Now, we set up and evaluate the integral for the volume: \[ V = \pi \int_{0}^{4} [(4x)^2 - (x^2)^2] dx \] \[ V = \pi \int_{0}^{4} (16x^2 - x^4) dx \] \[ V = \pi \left[ \frac{16x^3}{3} - \frac{x^5}{5} \right]_{0}^{4} \] \[ V = \pi \left[ \left(\frac{16(4)^3}{3} - \frac{(4)^5}{5}\right) - \left(\frac{16(0)^3}{3} - \frac{(0)^5}{5}\right) \right] \] \[ V = \pi \left[ \frac{16(64)}{3} - \frac{1024}{5} \right] \] \[ V = \pi \left[ \frac{1024}{3} - \frac{1024}{5} \right] \] \[ V = 1024\pi \left[ \frac{1}{3} - \frac{1}{5} \right] = 1024\pi \left[ \frac{5 - 3}{15} \right] = 1024\pi \left[ \frac{2}{15} \right] \] \[ V = \frac{2048\pi}{15} \]
Step 4: Final Answer:
The volume of the solid is \(\frac{2048\pi}{15}\) cubic units.