Problem 4 - Thomas Calculus

Question

Find the volumes of the solids. The solid lies between planes perpendicular to the \(x\) -axis at \(x=-1\) and \(x=1 .\) The cross-sections perpendicular to the \(x\) -axis between these planes are squares whose diagonals run from the semicircle \(y=-\sqrt{1-x^{2}}\) to the semicircle \(y=\sqrt{1-x^{2}}.\)

Step-by-Step Solution

Answer

The volume of the solid is \(\frac{16}{3}\).

Step 1: Understand the Problem

We need to find the volume of a solid whose cross-sections are squares. These squares have diagonals running from the lower semicircle \(y=-\sqrt{1-x^2}\) to the upper semicircle \(y=\sqrt{1-x^2}\). The solid is bounded between \(x=-1\) and \(x=1\) on the \(x\)-axis.

Step 2: Find the Length of the Diagonal

For a given \(x\), the diagonal of the square runs from \(y=-\sqrt{1-x^2}\) to \(y=\sqrt{1-x^2}\). The length of the diagonal is twice the value of \(y\) at that point, i.e., \(2\sqrt{1-x^2}\).

Step 3: Determine the Side of the Square

If \(d\) is the length of the diagonal of a square, the side length \(s\) can be found using the formula: \(s = \frac{d}{\sqrt{2}}\). Thus, the side length of the square is \(\frac{2\sqrt{1-x^2}}{\sqrt{2}} = \sqrt{2}\sqrt{1-x^2} = \sqrt{2(1-x^2)}\).

Step 4: Express the Area of the Square

The area \(A\) of a square with side \(s\) is given by \(s^2\). Therefore, \(A = (\sqrt{2(1-x^2)})^2 = 2(1-x^2)\).

Step 5: Set up the Integral for the Volume

To find the total volume, integrate the area of the cross-sectional square from \(x = -1\) to \(x = 1\). Thus, the volume \(V\) is \(V = \int_{-1}^{1} 2(1-x^2) \, dx\).

Step 6: Solve the Integral

Calculate the integral: \(\int_{-1}^{1} 2(1-x^2) \, dx = 2\left[\int_{-1}^{1} (1-x^2) \, dx\right]\). This breaks down to \(2\left[\int_{-1}^{1} 1 \, dx - \int_{-1}^{1} x^2 \, dx\right]\). The first integral is \(2[x]_{-1}^{1} = 2[1 - (-1)] = 4\). The second integral is \(2\left[\frac{x^3}{3}\right]_{-1}^{1} = 2\left[\frac{1^3}{3} - \frac{(-1)^3}{3}\right] = \frac{4}{3}\). Thus, the volume is \(2(4 - \frac{4}{3}) = 2\left(\frac{12}{3} - \frac{4}{3}\right) = 2\left(\frac{8}{3}\right) = \frac{16}{3}\).

Key Concepts

Cross-sections

When we talk about cross-sections in the context of finding volumes of solids, think of slicing a loaf of bread into slices of any shape or size. Each slice is a cross-section of the loaf. In this particular problem, the solid is sliced perpendicular to the x-axis, and each of these slices is a square.

• The cross-section is not a random square; its diagonal stretches from one semicircle to another.
• The semicircles are mathematically defined by the functions \(y = \pm \sqrt{1-x^2}\), forming semi-circular caps on a circle centered at the origin.

This scenario creates a square where the diagonal is dependent on the value of \(x\). So, as \(x\) changes from -1 to 1, the shape of the square moves and changes too. Understanding the concept of cross-sections is crucial for visualizing how the volume of the solid is constructed as a sum of these infinitely many squares.

Integral Calculus

Integral calculus allows us to compute the total volume of the solid by adding up infinitely small elements, in this case, the areas of each square cross-section. As we set up our integral, we determine the contribution of each tiny slice within the intervals.

• The integral bounds are from \(x = -1\) to \(x = 1\), meaning we calculate the volume enclosed between these vertical planes.
• Each infinitesimally thin slice perpendicular to the x-axis is described by a function \(2(1-x^2)\), which now becomes the integrand.

To find the total volume of such slices (or squares), we sum these tiny changes from one end to the other using the integral \[ V = \int_{-1}^{1} 2(1-x^2) \, dx \].
This step turns a daunting volume calculation into a solvable mathematical expression. Integrating this function combines the calculated areas of all slices from -1 to 1, thus giving the volume of the solid.

Area of Squares

Understanding the area of squares when applied to solving volume problems involves grasping the relationship between different square parameters. Knowing one attribute, such as the diagonal, can help us find out others, like the side length and the area.

• For a square, the area \(A\) is calculated as the square of its side length \(s\), given by \(A = s^2\).
• Finding the side from the diagonal uses the formula \(s = \frac{d}{\sqrt{2}}\), which is a key step in this exercise.
• Thus, for a diagonal \(d = 2\sqrt{1-x^2}\), the side becomes \(\sqrt{2(1-x^2)}\), leading to an area formula of \(2(1-x^2)\).

The logic used here leverages geometric properties: the diagonal of a square reveals its side, and from the side, we can compute the area, ensuring every step is interconnected and consistent. This detailed understanding aids in preparing the integral necessary to find the volume of a more complex shape comprised of these cross-sectional squares.