Problem 36 - Essential Calculus

Question

Determine which value best approximates the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.) \(y=\arctan x, \quad y=0, \quad x=0, \quad x=1\) (a) 10 (b) \(\frac{3}{4}\) (c) 5 (d) -6 (e) 15

Step-by-Step Solution

Answer

(b) \(\frac{3}{4}\) is the most reasonable answer, as it is the only option which is less than \(\frac{1}{3}\pi\), roughly equal to \(1.05\).

Step 1: Sketch the Boundaries

Sketch the boundaries of the solid by drawing the graph of \(y = \arctan x\) between \(x = 0\) and \(x = 1\). Include also the lines \(x = 0\), \(x = 1\), and \(y = 0\) in the graph.

Step 2: Visualize the Solid

Envision the solid which is generated by revolving the region enclosed by the drawn lines about the x-axis. The shape should resemble a cone, with the base radius equal to \(\arctan 1\) or the maximum value of \(y\) from Step 1, and a height of \(1\) (from \(x = 0\) to \(x = 1\).

Step 3: Estimate the Volume

Remember the formula for the volume of a cone \(V = \frac{1}{3}\pi r^2h\), where \(r\) is the radius and \(h\) is the height. The height is \(1\) and radius is less than \(1\), so the volume of our solid of revolution should be less than \(\frac{1}{3}\pi\). Inspect the given options and find the one closest to this estimate.

Key Concepts

Volume Approximation

Volume approximation is a useful technique in calculus when working with solids of revolution. Instead of calculating the exact volume, we focus on estimating it. A solid of revolution is created when a region on a graph is spun around an axis, creating a 3D shape.
To approximate the volume of a solid of revolution:

• Sketch the region being revolved to grasp the visual shape of the solid.
• Break down the solid into simpler shapes, like cones or cylinders, which are easier to handle.
• Use geometric formulas, like those for the volume of cones or cylinders, to estimate the volume.

For the problem in question, by sketching, envisioning, and using the cone volume formula, we aimed to find the best approximate volume without detailed calculations. Sketching helps assess whether the estimated volume is realistic based on visual analysis.

Arctan Function

The arctan function, denoted as \(y = \arctan x\), is a crucial concept in trigonometry and calculus. It's the inverse of the tangent function, which helps in solving various mathematical problems, especially when dealing with angles and slopes. In the problem's context, \(y = \arctan x\) is part of the boundary of the region being rotated. This function's graph is a curve that starts at the origin, \(x = 0\), and moves upward toward \(x = 1\), creating a boundary for the solid.

• At \(x = 0\), \(\arctan 0 = 0\).
• At \(x = 1\), \(\arctan 1 \approx \frac{\pi}{4}\), marking the tallest point on the curve within the specified range.

Understanding this function helps predict the shape and dimensions of the revolution solid and, ultimately, estimate its volume.

Graph Sketching

Graph sketching is the skill of drawing a curve to understand the behavior and relationships of functions visually. This task is fundamental when dealing with solids of revolution. In this problem, sketching allows us to:

• Identify the boundaries and shapes involved in the revolution, such as the curve \(y = \arctan x\) and lines \(y = 0\), \(x = 0\), \(x = 1\).
• Visualize how the region bounded by these lines interacts and forms a solid when revolved around the \(x\)-axis.

When sketching, remember to:

• Highlight key intersections and points of interest, such as where the curve meets the axes.
• Ensure the scale is consistent to avoid misleading interpretations of the solid's size or shape.

Mastering graph sketching can simplify complex calculus concepts into easily digestible visuals, aiding in problem-solving.

Cone Volume Formula

The volume of a cone is a basic geometrical concept that plays a significant role in solving the given exercise. The formula is \(V = \frac{1}{3}\pi r^2h\), where \(r\) is the radius and \(h\) is the height. For the solid of revolution:

• Height \(h\) is the difference in the \(x\)-values, \(x=1 - x=0 = 1\).
• Radius \(r\) is determined by the maximum \(y\)-value, which is \(-\arctan x\) at \(x=1\).

Substituting these into the formula gives a straightforward way to approximate the volume. When comparing the result to options provided, one can identify the most accurate approximation by simply evaluating if the calculated volume is realistic compared to sketches and expectations. Utilizing this formula effectively allows for more efficient and accurate volume approximation in numerous mathematical problems.