Problem 67 - Calculus Single Variable

Question

In each of Exercises 65-68, use the method of cylindrical shells to calculate the volume obtained by rotating the given planar region \(\mathcal{R}\) about the given line \(\ell\) \(\mathcal{R}\) is the region between the curves \(y=x^{2}-x-5\) and \(y=-x^{2}+x+7 ; \ell\) is the line \(x=5\)

Step-by-Step Solution

Answer

The volume is \(523\pi\).

Step 1: Determine the Intersection Points

First, we need to find the points of intersection for the curves to determine the bounds for integration. Set the equations equal to each other: \(x^2 - x - 5 = -x^2 + x + 7\). Simplify to get: \(2x^2 - 2x - 12 = 0\). Divide by 2: \(x^2 - x - 6 = 0\). Using the quadratic formula, solve \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 1\), \(b = -1\), and \(c = -6\). This gives \(x = \frac{1 \pm 5}{2}\), resulting in \(x = 3\) and \(x = -2\).

Step 2: Set Up the Integral for Volume Using Cylindrical Shells

Using the method of cylindrical shells, the volume \(V\) is given by the integral \[ V = \int_{a}^{b} 2\pi (R - x)(f(x) - g(x)) \, dx \] where \(R = 5\) (the line about which we are rotating), \(f(x) = -x^2 + x + 7\), and \(g(x) = x^2 - x - 5\).

Step 3: Compute the Integral Bounds and Functions

Determine the expression inside the integral. The function inside becomes: \(f(x) - g(x) = [-x^2 + x + 7] - [x^2 - x - 5] = -2x^2 + 2x + 12\). The distance from the line \(x=5\) is \(R - x = 5 - x\). Thus, the volume integral is \[ V = \int_{-2}^{3} 2\pi (5-x)(-2x^2 + 2x + 12) \, dx \].

Step 4: Evaluate the Integral

Expand the expression: \((5-x)(-2x^2 + 2x + 12)\) as \(-10x^2 + 10x + 60 + 2x^3 - 2x^2 - 12x\), which simplifies to \(2x^3 - 12x^2 - 2x + 60\). The integral is now \[ V = 2\pi \int_{-2}^{3} (2x^3 - 12x^2 - 2x + 60) \, dx \].Calculate the integral of each term:1. \(2x^3 \to \frac{x^4}{2}\)2. \(-12x^2 \to -4x^3\)3. \(-2x \to x^2\)4. \(60 \to 60x\)The integral from \(-2\) to \(3\) is then calculated, resulting in a specific numeric value.

Step 5: Compute the Definite Integral and Final Volume

Evaluate each term of the integral between the bounds \(-2\) and \(3\):1. \(1/2 \times (3)^4 - 1/2 \times (-2)^4 = 40.5 - 8 = 32.5\)2. \(-4 \times (3)^3 + 4 \times (-2)^3 = -108 + 32 = -76\)3. \((3)^2 - (-2)^2 = 9 - 4 = 5\)4. \(60 \times 3 - 60 \times (-2) = 180 + 120 = 300\)Combine and multiply by \(2\pi\): \[ 2\pi(32.5 - 76 + 5 + 300) = 2\pi \times 261.5 = 523\pi \].

Key Concepts

Volume of Solids of Revolution

The concept of finding the volume of solids of revolution revolves around rotating a region or shape around a line to create a three-dimensional object. In this case, we are using the method of cylindrical shells, a technique particularly useful when dealing with vertical lines of rotation.

Imagine taking a small section—or shell—from the object you're rotating. The volume of that shell contributes to the total volume, and you achieve this by performing an integration process. It's akin to stacking up several cylindrical layers to form a 3D solid. Each of these layers has a small thickness, and the collection of these layers in the direction of rotation results in the entire solid.

The volume of each cylindrical shell is determined using the formula for shells:

• Height of the shell: Given by the difference in function values, \(f(x) - g(x)\).
• Distance from the axis: \(R - x\), where \(R\) is the line of rotation.

The integral of these values, over the interval of intersection, yields the total volume.

Intersection of Curves

Finding the intersection of curves is crucial in determining the bounds for integration when calculating the volume of solids of revolution. Given the graphs of two curves, the intersection points are where the curves meet, and these points define the integral's limits.

To find the intersection, you equate the two functions. For instance, with curves provided by functions \(y = x^2 - x - 5\) and \(y = -x^2 + x + 7\), we solve \(x^2 - x - 5 = -x^2 + x + 7\). Through simplification, we form a quadratic equation: \(x^2 - x - 6 = 0\). Solving this equation gives the \(x\)-coordinates of the intersection points.

These are the points where both curves cross each other, defining the region you will rotate.

Definite Integral

The definite integral is a key tool in calculus to calculate the exact volume of a solid between specified bounds. It essentially sums up an infinite number of tiny values over a continuous interval, providing the accumulation needed to find areas, volumes, or various physical quantities.

In this exercise, the definite integral is used to determine the volume of the solid obtained by rotating the region. By setting up the integral \[ V = \int_{a}^{b} 2\pi (R - x)(f(x) - g(x)) \, dx \]we integrate from \(a = -2\) to \(b = 3\), which are intersection points on the \(x\)-axis.

The result of this integral gives us the total volume when multiplied by \(2\pi\). Definite integrals bring together the infinite process of summation in a specific range indicated by these bounds.

Quadratic Formula

The quadratic formula is a method for finding the roots of a quadratic equation, which is any equation that can be expressed in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. This formula is particularly helpful to find points of intersection or solve problems involving quadratics.

The formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \(b^2 - 4ac\) is the discriminant. The roots provided by this formula tell us the points where the curve intersects the \(x\)-axis or, in this exercise, where two curves intersect each other.

For the equation \(x^2 - x - 6\), applying the quadratic formula reveals two solutions: \(x = 3\) and \(x = -2\). These solutions are critical as they define the interval over which the definite integral for the cylindrical shell method is calculated.