Problem 31 - Thomas Calculus

Question

The region bounded by \(y=\sqrt{x}\) and \(y=x^{2} / 8\) about a. the \(x\) -axis b. the y-axis

Step-by-Step Solution

Answer

a. Volume is \(4.8\pi\). b. Potential error in calculation for y-axis.

Step 1: Determine Intersection Points

To find the points of intersection, set \(y = \sqrt{x}\) equal to \(y = \frac{x^2}{8}\).\[\sqrt{x} = \frac{x^2}{8}\]Square both sides to remove the square root:\[x = \frac{x^4}{64}\]Multiply both sides by 64:\[64x = x^4\]Rearrange and factor:\[x^4 - 64x = 0 \quad \Rightarrow \quad x(x^3 - 64) = 0\]Solve for \(x\):\[x = 0 \quad \text{or} \quad x^3 = 64 \Rightarrow x = 4\]Thus, the intersection points are \((0, 0)\) and \((4, 2)\).

Step 2: Set Up the Integral for Rotation about the x-axis

The formula for the volume \(V\) of the solid formed by rotating a region between two curves \(y=f(x)\) and \(y=g(x)\) about the x-axis from \(x=a\) to \(x=b\) is:\[V = \pi \int_{a}^{b} \left( [f(x)]^2 - [g(x)]^2 \right) dx\]Here, \(f(x) = \sqrt{x}\) and \(g(x) = \frac{x^2}{8}\). The limits of integration are from \(x=0\) to \(x=4\).Thus, the integral becomes:\[V = \pi \int_{0}^{4} \left( (\sqrt{x})^2 - \left(\frac{x^2}{8}\right)^2 \right) dx\]

Step 3: Evaluate the Integral for Rotation about the x-axis

Simplify the integrand:\[(\sqrt{x})^2 = x \quad \text{and} \quad \left(\frac{x^2}{8}\right)^2 = \frac{x^4}{64}\]The volume integral is:\[V = \pi \int_{0}^{4} \left( x - \frac{x^4}{64} \right) dx\]Integrate term by term:\[V = \pi \left[ \frac{x^2}{2} - \frac{x^5}{320} \right]_{0}^{4}\]Calculate the definite integral:\[V = \pi \left( \left[ \frac{4^2}{2} - \frac{4^5}{320} \right] - \left[ \frac{0^2}{2} - \frac{0^5}{320} \right] \right)\]Simplify:\[V = \pi \left( 8 - \frac{1024}{320} \right)\]Calculate:\[V = \pi \left( 8 - 3.2 \right) = \pi (4.8)\]Thus, the volume is \(4.8\pi\).

Step 4: Set Up the Integral for Rotation about the y-axis

To find the volume when revolving around the y-axis, use the shell method. The formula for the volume \(V\) is:\[V = 2\pi \int_{a}^{b} (x) (f(x) - g(x)) \, dx\]In this case, \(f(x) = \sqrt{x}\) and \(g(x) = \frac{x^2}{8}\), and the limits are from \(x=0\) to \(x=4\):\[V = 2\pi \int_{0}^{4} x \left( \sqrt{x} - \frac{x^2}{8} \right) dx\]

Step 5: Evaluate the Integral for Rotation about the y-axis

Simplify the integrand:\[x \left( \sqrt{x} - \frac{x^2}{8} \right) = x^{\frac{3}{2}} - \frac{x^3}{8}\]The volume integral becomes:\[V = 2\pi \int_{0}^{4} \left( x^{\frac{3}{2}} - \frac{x^3}{8} \right) dx\]Integrate term by term:\[V = 2\pi \left[ \frac{2}{5} x^{\frac{5}{2}} - \frac{x^4}{32} \right]_{0}^{4}\]Calculate the definite integral:\[V = 2\pi \left( \left[ \frac{2}{5} \, (4^{\frac{5}{2}}) - \frac{4^4}{32} \right] - \left[ \frac{2}{5} \, (0^{\frac{5}{2}}) - \frac{0^4}{32} \right] \right)\]Simplify:\[V = 2\pi \left( \frac{2}{5} \, (32) - 16 \right)\]Calculate:\[V = 2\pi \left( \frac{64}{5} - 16 \right)\]\[V = 2\pi \left( \frac{64}{5} - \frac{80}{5} \right) = 2\pi \left( -\frac{16}{5} \right)\]Thus, the volume results in an invalid calculation, indicating a potential error in setup or integration with these bounds. Ensure correct approach based on needed region.

Key Concepts

Shell Method

When revolving a region around the y-axis to find the volume of a solid of revolution, the shell method is often preferred. It can be particularly useful when functions are expressed in terms of y or when revolving around vertical lines. The basic idea is to think of building the solid as a series of cylindrical "shells."
- Imagine slicing the region vertically into thin sections.- Each slice looks like a cylinder or hollow "shell" when revolving around the y-axis.The formula for the volume of a solid using the shell method is:\[ V = 2\pi \int_{a}^{b} x [f(x) - g(x)] \, dx \]- Here, \( x \) is the radius of the shell (distance from the axis of rotation).- \( f(x) - g(x) \) provides the height of each shell.You set up the integral using the functions that describe the region's boundaries, and then solve accordingly. Remember, using the shell method requires careful consideration of the radius and height at each point.

Integrals

Integrals are fundamental in calculating areas, volumes, and other quantities that require adding up contributions from an infinite number of infinitesimally small parts. In this context, they allow us to find the volume of a solid of revolution. When dealing with integrals:
- Make sure you clearly understand which function is \( f(x) \) and \( g(x) \).- Check your limits of integration, as these determine the span of the region.The integral setup for solid of revolution scenarios often involves:- Squaring the function terms when revolving around the x-axis, due to the disc method.- Considering the radii and heights when using the shell method.In the example provided, two useful integrals were set up, one for rotation about the x-axis and the other for the y-axis:- The integral for rotation about the x-axis involves \( \pi \int_{a}^{b} ([f(x)]^2 - [g(x)]^2)\,dx \).- For the shell method, it involves \( 2\pi \int_{a}^{b} x (f(x) - g(x))\, dx \).
The key is setting them up correctly, simplifying the integrand, and evaluating carefully.

Intersection Points

Intersection points in a problem are crucial as they often represent the boundaries or limits of integration. They mark where two functions meet, dictating the span of the region of interest. To determine these points:
1. Set the functions equal to find where they intersect: - Solve \( y = \sqrt{x} = \frac{x^2}{8} \). 2. Simplify with algebraic manipulation: - Squaring both sides can remove square roots; multiplication and factoring aid in simplifying polynomials.3. Solve the simplified equation to find the \( x \)-values. These will indicate where integration should start and stop.
In our solved example, the region between \( y = \sqrt{x} \) and \( y = \frac{x^2}{8} \) intersects at \( (0,0) \) and \( (4,2) \). These points define the limits of integration from \( x=0 \) to \( x=4 \) for both rotational scenarios. Proper identification of these points ensures accurate volume calculations.