Consider the region R in the first quadrant bounded by y=x^1/n and y=x^n, where n>1 is a positive number.


a. Find the volume V(n) of the solid generated when R is revolved about the x-axis. Express your answer in terms of n.

Verified step by step guidance

1. Identify the region R bounded by the curves \(y = x^{\frac{1}{n}}\) and \(y = x^n\) in the first quadrant. Since \(n > 1\), these two curves intersect at \(x=0\) and \(x=1\) because \(x^{\frac{1}{n}} = x^n\) implies \(x=0\) or \(x=1\).
2. Set up the volume integral using the method of washers (disks with holes) when revolving around the x-axis. The volume \(V(n)\) is given by integrating the difference of the squares of the outer and inner radii:
3. \[V(n) = \pi \int_0^1 \left[ (\text{outer radius})^2 - (\text{inner radius})^2 \right] \, dx.\]
4. Determine which curve is the outer radius and which is the inner radius with respect to the x-axis. Since \(y = x^{\frac{1}{n}}\) is greater than \(y = x^n\) on \((0,1)\) for \(n > 1\), the outer radius is \(x^{\frac{1}{n}}\) and the inner radius is \(x^n\).
5. Write the integral explicitly as: \[V(n) = \pi \int_0^1 \left[ (x^{\frac{1}{n}})^2 - (x^n)^2 \right] \, dx = \pi \int_0^1 \left( x^{\frac{2}{n}} - x^{2n} \right) \, dx.\] Then, integrate term-by-term using the power rule for integrals.

Key Concepts

Defining the Region Bounded by Curves

Understanding the region R requires identifying the curves y = x^(1/n) and y = x^n and their intersection points in the first quadrant. Since n > 1, these functions define a closed area between them for x in [0,1]. Recognizing the limits of integration is essential for setting up the volume integral.

Volume of Solids of Revolution Using the Disk/Washer Method

When a region is revolved about the x-axis, the volume can be found by integrating the area of circular cross-sections perpendicular to the axis. The washer method applies when there is an inner and outer radius, calculated from the bounding curves, and the volume is the integral of π(outer radius² - inner radius²) dx over the interval.

Integration of Power Functions

The volume integral involves integrating expressions like x^(2/n) and x^(2n), which are power functions. Applying the power rule for integration, ∫x^m dx = x^(m+1)/(m+1), is necessary to find a closed-form expression for V(n) in terms of n.