Problem 106 - Calculus Volume 1

Question

A better approximation of the volume of a football is given by the solid that comes from rotating \(y=\sin x\) around the \(x\) -axis from \(x=0\) to \(x=\pi .\) What is the volume of this football approximation, as seen here?

Step-by-Step Solution

Answer

The volume of the approximated football is \( \frac{\pi^2}{2} \).

Step 1: Understand the Problem

The goal is to find the volume of a solid generated by rotating the function \( y = \sin x \) around the \(x\)-axis from \( x = 0 \) to \( x = \pi \). This forms a three-dimensional shape.

Step 2: Formula for Volume of Revolution

When a curve \( y = f(x) \) is rotated around the \(x\)-axis, the volume \( V \) of the solid formed is given by the integral \( V = \int_a^b \pi (f(x))^2 \, dx \). Here, \( f(x) = \sin x \), \( a = 0 \), and \( b = \pi \).

Step 3: Set Up the Integral

We will substitute \( f(x) = \sin x \) into the volume formula to get \( V = \int_0^\pi \pi (\sin x)^2 \, dx \).

Step 4: Simplify the Integral

To simplify the integral \( \int_0^\pi \pi (\sin x)^2 \, dx \), we use the identity \( \sin^2 x = \frac{1 - \cos(2x)}{2} \). This gives us the integral \( V = \int_0^\pi \pi \frac{1 - \cos(2x)}{2} \, dx \).

Step 5: Evaluate the Integral

Separate the integral into two parts: \( V = \frac{\pi}{2} \left[ \int_0^\pi 1 \, dx - \int_0^\pi \cos(2x) \, dx \right] \). Evaluate these integrals: \( \int_0^\pi 1 \, dx = \pi \) and \( \int_0^\pi \cos(2x) \, dx = \left. \frac{1}{2} \sin(2x) \right|_0^\pi = 0 \).

Step 6: Calculate the Final Volume

Plugging the results back in gives \( V = \frac{\pi}{2} [\pi - 0] = \frac{\pi^2}{2} \). Hence, the volume of the rotated solid is \( \frac{\pi^2}{2} \).

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics that deals with finding the total accumulation of quantities. In our context, it's used to compute the volume of solids generated by a curve when it's rotated around an axis. This volumetric computation is done via integration. The basic idea is to sum up infinite infinitely small slices to find the total volume.

Here's how it applies specifically:

• For a function like \( y = f(x) \), rotating it around the \( x \)-axis involves integrating \( \pi (f(x))^2 \, dx \) over a given interval \([a, b]\).

Since the integration stretches across a continuous interval, it captures all tiny contributions of volume along the axis of rotation. This is a foundational technique in integral calculus, especially useful in physics and engineering applications.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, represent relationships in a right angle triangle. In this exercise, we use the sine function, represented by \( y = \sin x \). This wave-like function describes a simple harmonic motion, regularly oscillating between -1 and 1.

When using \( y = \sin x \) for volume calculation:

• The function's oscillating value determines the height of each infinitesimal disk that forms the solid of revolution.
• An important identity used in calculus involving \( \sin^2 x \) is \( \sin^2 x = \frac{1 - \cos(2x)}{2} \). This simplifies calculations during integration.

Integrating this function over a specified range can represent physical phenomena, such as wave patterns or oscillations, which are common in nature and various technologies.

Solid of Revolution

A solid of revolution is a three-dimensional shape generated by rotating a two-dimensional curve around an axis. This technique helps visualize real-world objects' volumes, like vases, basins, or, in our case, a football.

Here's how it works:

• By choosing a curve (e.g., \( y = \sin x \)) and a specified interval, infinite tiny disks (or washers) are stacked together, whose volumes combine to form the solid.
• The formula \( V = \int_a^b \pi (f(x))^2 dx \) uses \( \pi \) due to the rotation around the \( x \)-axis, incorporating the circular geometry of each disk.
• The technique requires careful selection of limits – beginning and ending values for \( x \) – which define the solid's boundaries.

This method is essential in both theoretical studies and practical engineering designs, offering a model to calculate complex shapes' volumes with high precision.