Problem 3 - 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 bases 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 8 cubic units.

Step 1: Understanding the Problem

We're given shapes (squares) with bases on a semicircle from \(y = -\sqrt{1-x^2}\) to \(y = \sqrt{1-x^2}\) for \(x\) values from \(-1\) to \(1\). We need to find the volume of the 3D solid formed by these squares as they move along the \(x\)-axis.

Step 2: Determine the Side Length of the Square

Each square is positioned with its base from \(y = -\sqrt{1-x^2}\) to \(y = \sqrt{1-x^2}\), making the base length \(\text{side} = 2\sqrt{1-x^2}\).

Step 3: Express the Area of the Cross-Section

Since the cross-section is a square, the area \(A(x)\) can be calculated as the square of its side length. Therefore, \(A(x) = \left(2\sqrt{1-x^2}\right)^2 = 4(1-x^2)\).

Step 4: Setup the Integral for the Volume

We need the volume of the solid, which is obtained by integrating the area of the cross-section from \(x = -1\) to \(x = 1\): \[ V = \int_{-1}^{1} A(x)\, dx = \int_{-1}^{1} 4(1-x^2)\, dx \]

Step 5: Evaluate the Integral

Calculate the definite integral for volume:\[V = \int_{-1}^{1} 4(1-x^2)\, dx = 4 \left[ x - \frac{x^3}{3} \right]_{-1}^{1} \]First evaluate at \(x = 1\):\(4 \left( 1 - \frac{1^3}{3} \right) = 4 \left( 1 - \frac{1}{3} \right) = \frac{4 \cdot 2}{3} = \frac{8}{3}\)Then at \(x = -1\):\(4 \left( -1 + \frac{(-1)^3}{3} \right) = 4 \left( -1 + \frac{-1}{3} \right) = 4 \left( -\frac{4}{3} \right) = -\frac{16}{3}\)Volume calculation combining both results:\(V = \frac{8}{3} - \left(-\frac{16}{3}\right) = \frac{8}{3} + \frac{16}{3} = \frac{24}{3} = 8\)

Step 6: Final Volume Calculation

The integral has been evaluated, giving us a final volume. Therefore, the volume of the solid is 8 cubic units.

Key Concepts

Cross-Sectional Area

Cross-sectional areas are key to understanding the structure of a solid. When imagining a 3D solid, visualize cutting it down into simpler 2D shapes, called cross-sections. This process eases the calculation of volume.
In our exercise, the solid is sliced into square cross-sections along the x-axis. The base of each square stretches from the lower bound of the semicircle, given by the equation \(y = -\sqrt{1-x^2}\), to the upper bound \(y = \sqrt{1-x^2}\). This means that as we move along different values of \(x\), the length of each side of these square cross-sections will change.

• The base runs between two points: \(-\sqrt{1-x^2}\) and \(\sqrt{1-x^2}\).
• The length of the base is calculated as \(2\sqrt{1-x^2}\).

For a square, the area is the side length squared. Therefore, each cross-sectional area \(A(x)\) becomes \((2\sqrt{1-x^2})^2 = 4(1-x^2)\).

Definite Integral

The process of finding the volume of a solid using integration involves setting up and evaluating a definite integral. The definite integral allows you to add up the infinite cross-sectional areas along the interval where the solid exists.
When you compute a definite integral, you essentially sum up tiny bits, or increments, of area. These are represented as small slices along the \(x\)-axis, and by adding these slices together, you approximate the entire volume of the solid.

• The definite integral from our exercise: \[ V = \int_{-1}^{1} 4(1-x^2)\, dx \] calculates the total volume.
• It is evaluated between \(x = -1\) and \(x = 1\), enclosing the full span where the cross-sections are defined.

By computing this, you model the 3D solid perfectly, capturing every possible slice. Solving this integral, we find the volume of the 3D shape to be 8 cubic units.

Solid of Revolution

A solid of revolution is a 3D object created by rotating a 2D shape around an axis. It's similar to creating pottery on a spinning wheel—the shape you draw creates the full volumetric form as it spins.
For the problem in question, however, instead of revolving a shape, we use cross-sectional areas orthogonal (at a right angle) to the axis, which is somewhat similar. In this case, squares laid along the \(x\)-axis are similar in principle, because they're symmetric around the central axis of the semicircle.

• The semicircle equation, \(y = \pm\sqrt{1-x^2}\), creates a set boundary for our cross-sections.
• Though it's not a traditional 'revolution', the formation mirrors the idea since the final solid maintains symmetry.

Aspiring to carve out curved shapes, understanding solids of revolution helps us appreciate the complexity and symmetric beauty of such geometric constructs. It connects planar geometry to volumetric calculations, making the leap from 2D to 3D understanding much more intuitive.