Problem 73 - Calculus Volume 1

Question

For the following exercises, draw an outline of the solid and find the volume using the slicing method. The base is the area between \(y=x\) and \(y=x^{2}\) . Slices perpendicular to the \(x\) -axis are semicircles.

Step-by-Step Solution

Answer

The volume of the solid is \(\frac{\pi}{240}\).

Step 1: Understanding the Region

We need to visualize the area between the curves \(y = x\) and \(y = x^2\). These two curves intersect at points where \(x = x^2\), which means \(x^2 - x = 0\), giving \(x(x - 1) = 0\). Thus, the intersection points are \(x = 0\) and \(x = 1 \). The region of interest is "triangular" shaped bounded by these two curves from \(x = 0\) to \(x = 1\).

Step 2: Establishing the Slices

The problem specifies that slices perpendicular to the \(x\)-axis are semicircles. For a given \(x\), the diameter of each semicircle is the vertical distance between the two curves: \(y = x - x^2\). The radius \(r\) of a semicircle is half of the diameter, \(r = \frac{x - x^2}{2}\).

Step 3: Calculate the Area of a Semicircle

The area \(A(x)\) of a semicircle with radius \(r\) is \(A(x) = \frac{1}{2}\pi r^2\). Substituting \(r\) with \(\frac{x - x^2}{2}\), we have: \[ A(x) = \frac{1}{2}\pi \left(\frac{x - x^2}{2}\right)^2 = \frac{\pi}{8}(x - x^2)^2. \]

Step 4: Integrate to Find Volume

To find the volume \(V\) of the solid, integrate the area \(A(x)\) from \(x = 0\) to \(x = 1\): \[ V = \int_{0}^{1}\frac{\pi}{8}(x - x^2)^2\,dx. \] This requires expansion of \((x - x^2)^2\) to \(x^2 - 2x^3 + x^4\) and then integration term by term: \[ \int_{0}^{1}\frac{\pi}{8}(x^2 - 2x^3 + x^4)\,dx = \frac{\pi}{8} \left[ \frac{x^3}{3} - \frac{x^4}{2} + \frac{x^5}{5} \right]_{0}^{1}. \]

Step 5: Evaluate the Integral

Evaluate the definite integral: \[ \frac{\pi}{8} \left( \frac{1^3}{3} - \frac{1^4}{2} + \frac{1^5}{5} \right) = \frac{\pi}{8} \left( \frac{1}{3} - \frac{1}{2} + \frac{1}{5} \right) = \frac{\pi}{8} \cdot \frac{10}{30} - \frac{15}{30} + \frac{6}{30} = \frac{\pi}{8} \cdot \frac{1}{30}. \]

Step 6: Final Calculation

Thus, the volume \(V\) of the solid is \(\frac{\pi}{240}\). This is the accumulated volume from integrating the semicircular areas over the region between \(x = 0\) and \(x = 1\).

Key Concepts

Slicing Method

The slicing method is a powerful technique in calculus used to find the volume of a solid. Imagine cutting an object into thin slices that are easier to analyze. By calculating the volume of each individual slice and summing them up, we can determine the total volume of the solid.

Here's how it works in practice: assume that the base of a solid lies on a particular plane, and imagine slicing this solid perpendicular to this plane. Each slice usually has a defined geometric shape, the area of which can be expressed in terms of a variable. **The key steps are**:

• Identify the base: Determine the limits of integration, which are the edges or boundaries of the base area.
• Establish slice shape: Determine how the slice is oriented, and its cross-sectional area in terms of the variable.
• Integrate: Use calculus to sum these infinitesimally thin slices over the given bounds.

When done correctly, the slicing method provides an accurate computation of volumes even for heavily irregular shapes.

Volume of Solids

Finding the volume of solids with irregular shapes can be a challenge. Calculus offers us tools to overcome these challenges. We specifically use integration to calculate the volume of solids when dealing with complex geometrical shapes.

**For example**: Consider a solid with a circular base or semicircular slices. The volume of such solids can be calculated by determining the cross-sectional area at each slice along one of the axes. By summing these areas, we find the volume, which involves integrating a function representing the area of the slice.

This is a general approach in calculus and can be applied to various shapes, whether polygons, curves or through rotational methods that form three-dimensional shapes from two-dimensional shapes.

Definite Integrals

In calculus, definite integrals are used for calculating the area under a curve, the net change in quantities, and in this context, the volume of a solid.

Definite integrals are expressed with set bounds, referred to as the 'limits of integration' that delineate the start and endpoints over which the function is integrated.

• The integral symbol \(int \) signifies the operation of integration.
• The bounds signify the limits over which you're integrating, such as from `x = a` to `x = b`.
• Functions are integrated to reflect the accumulation of volume, area, or other quantities.

Using definite integrals, computations of areas, and volumes become systematic and precise, and they are foundational in understanding how accumulative changes are quantified in mathematics.

Semicircular Slices

Semicircular slices refer to sections of a solid cut perpendicular to an axis, resembling half-circles. In calculus, these semicircular slices are significant for determining properties like volume. For example, if slices are made perpendicular to the x-axis, each may form a semicircle whose diameter depends on the intersecting curves.

**Key Steps**:

• Identify the diameter: It's the distance between intersecting curves.
• Find the radius: The radius \( r \) is half the diameter.
• Calculate area: Use the semicircle area formula, \( A(x) = \frac{1}{2} \pi r^2 \), which is derived from the full-circle area formula adjusted for semicircles.

By integrating the area of these semicircular slices across their defined bounds, we calculate the solid's volume efficiently and accurately. This is especially useful in problems with specific geometric constraints leading to a predictable pattern.