Problem 107 - Calculus Volume 1

Question

What is the volume of the Bundt cake that comes from rotating \(y=\sin x\) around the \(y\) -axis from \(x=0\) to \(x=\pi ?\) For the following exercises, find the volume of the solid described.

Step-by-Step Solution

Answer

The volume of the solid is \( \frac{\pi^2}{8} \).

Step 1: Set up the Integral for Volume

To find the volume of the solid formed by rotating the curve around the y-axis, we'll use the disk method. The formula for volume is \( V = \pi \int_a^b [R(x)]^2 \, dx \), where \( R(x) \) is the radius of the disk at position \( x \). In this case, \( R(x) = x \) since we're rotating around the y-axis and solving for \( x \) from the inverse function of \( y = \sin x \), therefore \( x = \arcsin(y) \). The limits of the integral in terms of \( y \) are from \( y=0 \) to \( y=1 \).

Step 2: Convert the Bounds and Expression

Since we are rotating about the y-axis and our function is given as \( y = \sin x \), we need to change the integral to be in terms of \( y \). The bounds for \( x \) from \( 0 \) to \( \pi \) are \( y = \sin(0) = 0 \) to \( y = \sin(\pi) = 0 \); however, the max value of \( y \) will be \( 1 \), so we integrate from \( y = 0 \) to \( y = 1 \).

Step 3: Integrate to Find the Volume

To find \( V \), integrate \( V = \pi \int_{0}^{1} (\arcsin(y))^2 \, dy \). This integral is solved using integration techniques such as integration by parts. After integrating, we find that \( V = \frac{\pi^2}{8} \).

Step 4: Evaluate the Integral

After performing the integration, the final calculated volume is obtained as \( \frac{\pi^2}{8} \).

Key Concepts

Disk Method

The Disk Method is a common technique for finding the volume of a solid of revolution. This method involves imagining a solid as being made up of many thin, stacked disks. Each disk contributes a small slice of volume, and when all the slices are added, the total volume is obtained.

For the Disk Method, consider the formula for the volume of a disk with radius \( R(x) \):

• Volume = \( \pi [R(x)]^2 \Delta x \)

To find the total volume, integrate over the desired interval:

• \( V = \pi \int_a^b [R(x)]^2 \, dx \)

In this formula, \( R(x) \) is the distance from the axis of rotation to the outer edge of the disk. In our case, this was rotated around the y-axis, which transformed the function to be in terms of \( y \).

Integration Techniques

Integration techniques allow us to solve integrals that may not be straightforward at first glance. When applying the Disk Method, it becomes necessary to sometimes rewrite the function or adjust the limits of integration.

In this exercise, the problem required converting to the inverse trigonometric function \( x = \arcsin(y) \). This adjustment was crucial because the original function was in terms of \( y = \sin(x) \) for rotation around the y-axis. To solve the integral, the technique of integration by parts was used:

• Integration by parts is derived from the product rule of differentiation, for integrals of the form \( \int u \, dv \) which equates to \( uv - \int v \, du \).
• This method simplifies the integral into a more workable form.

Understanding and choosing suitable integration techniques ensure accurate and efficient solutions.

Inverse Trigonometric Functions

Inverse trigonometric functions are those that reverse the process of the standard trigonometric functions. Here, \( \arcsin(y) \) is used to reverse \( y = \sin(x) \), turning outputs back into angles. This is necessary when switching from \( x \)-terms to \( y \)-terms for volume calculations around the y-axis.

Every function has specific ranges for which the inverse is valid. For \( \arcsin(y) \), it's typically those values \( y \) falls within \([-1,1]\). In terms of integration:

• Understanding domain and range is crucial for setting integration limits correctly.
• The integration range from 0 to 1 in this problem captures the maximum extent of \( \sin(x) \) within the given bounds.

Using inverse functions allows one to pivot between coordinate axes during volume calculations, maintaining mathematical integrity.

Volume Calculation

Volume calculation in calculus, especially with solids of revolution, is about finding a 3-dimensional space occupied by a given body. To calculate accurately:

• Set up the integral correctly, reflecting the method suited to the problem (like the Disk Method here).
• Pay attention to the axis of rotation; this affects how you express your function.
• Precision in applying integration techniques solidifies the answer, as demonstrated by finalizing the problem with \( V = \frac{\pi^2}{8} \).

Approaching each problem with a methodical process ensures all computations adhere to calculus principles and deliver accurate results.