Problem 45 - Thomas Calculus in SI Units

Question

Find the volume of the solid generated by revolving each region about the given axis. The region in the first quadrant bounded above by the curve \(y=x^{2},\) below by the \(x\) -axis, and on the right by the line \(x=1\) about the line \(x=-1\)

Step-by-Step Solution

Answer

The volume is \( \frac{7\pi}{6} \) cubic units.

Step 1: Visualize the Region and Setup the Problem

Begin by sketching the region in the first quadrant. The curve \( y = x^2 \) is a parabola opening upwards. The region of interest is bounded above by \( y = x^2 \), on the right by \( x = 1 \), and below by the \( x \)-axis. We will revolve this region about the line \( x = -1 \). This setup is important to visualize the shape of the solid formed by the revolution.

Step 2: Identify the Method of Integration

To find the volume of the solid, we will use the method of cylindrical shells. The formula for the volume using cylindrical shells when revolving around a vertical line \( x = c \) is:\[V = 2\pi \int_{a}^{b} (x-c)h(x)\,dx\]where \( h(x) \) is the height of the shell.

Step 3: Determine the Height and Radius of the Shells

For the region we are revolving, the height \( h(x) \) of each shell is given by the \( y \)-value of the curve, so \( h(x) = x^2 \). The radius of rotation is the distance from \( x \) to \( x = -1 \) which is \( x - (-1) = x + 1 \).

Step 4: Setup the Integral for the Volume

Using the cylindrical shells method, setup the integral as follows:\[V = 2\pi \int_{0}^{1} (x+1) x^2 \, dx\]This integral represents the sum of circumferences of the shells multiplied by their respective heights and thicknesses.

Step 5: Evaluate the Integral

Expand \((x+1)x^2\) before integrating:\[(x+1)x^2 = x^3 + x^2\]Now integrate term by term:\[V = 2\pi \int_{0}^{1} (x^3 + x^2) \, dx = 2\pi \left[ \frac{x^4}{4} + \frac{x^3}{3} \right]_{0}^{1}\]Evaluate the definite integral:\[V = 2\pi \left( \frac{1^4}{4} + \frac{1^3}{3} \right) = 2\pi \left( \frac{1}{4} + \frac{1}{3} \right)\]Combine the fractions:\[\frac{1}{4} + \frac{1}{3} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12}\]Finally, the volume is:\[V = 2\pi \cdot \frac{7}{12} = \frac{7\pi}{6}\]

Step 6: Summarize the Solution

The volume of the solid generated by revolving the region about the line \( x = -1 \) is \( \frac{7\pi}{6} \) cubic units.

Key Concepts

Cylindrical Shell Method

The Cylindrical Shell Method is a powerful technique in calculus for finding the volume of solids of revolution. This method utilizes the concept of cylindrical "shells" to sum up smaller volumes into a whole, very much like stacking rings of paper to form a tube. Instead of slicing the solid into discs or washers, the region is considered as being made of these subtractive, concentric cylindrical shells.

In our exercise, we revolve a region around an axis, specifically from a distance away from the axis of revolution. By choosing the cylindrical shell method, we essentially use vertical strips or shells to calculate the volume. The volume formula for a cylindrical shell is given by integrating over the range of interest, using:

• Height of the shell: Determined by the function itself, which gives the distance the shell extends in the vertical direction.
• Radius of the shell: The distance from the vertical axis of rotation, here defined by adding to the coordinate value.'x'.

Conceptually, the shells are easier to visualize when the axis of rotation is parallel to the strip being revolved, allowing us to exploit symmetry and simplify calculation processes extensively.

Definite Integrals

Definite integrals are key tools in calculus, especially useful for determining accumulated quantities such as area, volume, and total change. They extend the concept of antiderivatives and provide a precise mathematical means of calculating quantities with clear boundaries.

In solving our exercise, we employ the definite integral to evaluate the total volume delineated by revolving a region around an axis. The definite integral \(\int_{a}^{b} f(x) \, dx\) is used to integrate the function over the specific interval—framed by two boundaries where our shells span.

The integrals transform the idea of summing infinite small parts along the curve into a feasible and accurate volume calculation. Here, the integral sums up circumferences of varying length as the shell height alters along the parabolic boundary. Importantly, before computing, it's beneficial to manipulate the function to simplify the integration process, making it more streamlined and less prone to error during calculations.

Calculus Problem Solving

Calculus problem solving can often present challenges that require a blend of analytical thinking and strategic method selection. Here, the problem revolves around several key steps—visualization, method selection, and solution execution—all integral to mastering calculus.

Initially, visualizing the region being revolved is crucial. Sketch the curve, bounds, and rotational axis to conceive the 3D shape made by the rotation. This visualization aids in understanding which calculus method, among the several options, aligns best with solving the given problem. The Cylindrical Shell Method was chosen here due to its alignment with the vertical rotation and the nature of the bounded region.

Ultimately, problem solving in calculus is iterative and involves:

• Identifying pertinent details of the problem.
• Choosing and justifying the methods that simplify the process.
• Carefully executing mathematical operations such as integration and substitution.

Through practice and precision, these strategies refine your problem-solving skills and bolster your confidence in tackling a variety of calculus challenges.