Math Problem Statement

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the​ y-axis.

y equals xy=x​,

y equals 9 xy=9x​,

y equals 27y=27

y equals 9 xy=9x

y equals xy=x

y equals 27y=27

00

2727

Upper RR

xx

yy

Question content area bottom

Part 1

Set up the integral that gives the volume of the solid.

Integral from 0 to nothing left parenthesis nothing right parenthesis dy∫0enter your response hereenter your response here dy

​(Type exact​ answers.)

Solution

To solve this problem, we need to compute the volume of the solid generated by revolving the region RR about the yy-axis using the method of cylindrical shells or the method of disks/washers. Based on the equations provided:

The equations defining the region RR are:

  1. y=xy = x
  2. y=9xy = 9x
  3. y=27y = 27
  4. x=0x = 0 (this is implied since the region is bounded between x=0x = 0 and the points where the curves meet y=27y = 27).

We'll use the shell method since the region is being revolved around the yy-axis.

Step 1: Express xx in terms of yy.

  • For y=xy = x, this gives x=yx = y.
  • For y=9xy = 9x, this gives x=y9x = \frac{y}{9}.

Step 2: Set up the integral.

Using the method of cylindrical shells, the volume of a shell is given by:

V=y1y22πradiusheightdyV = \int_{y_1}^{y_2} 2\pi \cdot \text{radius} \cdot \text{height} \, dy

  • The radius of a shell is simply the xx-coordinate, which is x=y9x = \frac{y}{9} and x=yx = y.
  • The height of the shell is xrightxleftx_{\text{right}} - x_{\text{left}}, where xright=y9x_{\text{right}} = \frac{y}{9} and xleft=0x_{\text{left}} = 0.
  • The limits of integration will be from y=0y = 0 to y=27y = 27.

Thus, the volume VV is given by the integral:

V=2π027(yy9)dyV = 2\pi \int_0^{27} \left( y - \frac{y}{9} \right) \, dy

This is the setup for the integral. If you'd like, I can solve the integral next.

Would you like to proceed with solving the integral or need any further clarification?

Follow-up Questions:

  1. How do we apply the shell method when revolving around different axes?
  2. What is the difference between the shell method and the washer method?
  3. How do the limits of integration affect the volume calculation?
  4. Can the volume be calculated using the disk/washer method instead for this problem?
  5. How do transformations of curves change the region to be revolved?

Tip: When setting up volume integrals for revolved solids, always check if cylindrical shells or disks/washers provide a more straightforward approach based on the axis of revolution.

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Math Problem Analysis

Mathematical Concepts

Calculus
Volumes of Revolution
Shell Method

Formulas

V = 2π∫[a to b] (radius)(height) dy
x = y/9 for y = 9x
x = y for y = x

Theorems

Volume by Cylindrical Shells

Suitable Grade Level

Undergraduate Calculus (or Advanced High School Calculus)