General question about finding area of a solid by rotating axis

In summary, The question in the textbook is asking to sketch a graph and find the volume generated by revolving the region about the x-axis. The three equations given are y = x, y = 0, and x = 1. The solutions manual graphs y = x^3 and x + y = 10, which may be for a different edition. The original form of the text may not make sense, but the volumes considered could either be 0 or infinite. The text is likely talking about rotating the area contained by the three equations, which forms a right triangle with a vertex at the origin. By rotating this region, a cone with height 1 and radius 1 is generated.
  • #1
KataKoniK
1,347
0
I just have a general question here. There's this question in the textbook that asks to sketch a graph and then find the volume generated by revolving the region about the x-axis. The three equations they give is y = x, y = 0, x = 1.

Aren't we suppose to sketch a graph of those three equations? Because in the sols manual, they graph instead, y = x^3 and x + y = 10
 
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  • #2
KataKoniK said:
I just have a general question here. There's this question in the textbook that asks to sketch a graph and then find the volume generated by revolving the region about the x-axis. The three equations they give is y = x, y = 0, x = 1.

Aren't we suppose to sketch a graph of those three equations? Because in the sols manual, they graph instead, y = x^3 and x + y = 10
The solutions manual may be for a different edition, because there is no reason to graph the two equations it contains with respect to the original question.
 
  • #3
Thanks. I thought there was some mathematical reasoning behind graphing those two equations in the manual.
 
  • #4
KataKoniK said:
I just have a general question here. There's this question in the textbook that asks to sketch a graph and then find the volume generated by revolving the region about the x-axis. The three equations they give is y = x, y = 0, x = 1.

Aren't we suppose to sketch a graph of those three equations? Because in the sols manual, they graph instead, y = x^3 and x + y = 10

Nevermind the solution.In the original form,the text doesn't make any sense...The volumes considered are either 0 or infinite...

Daniel.
 
  • #5
Thanks Dan
 
  • #6
dextercioby said:
Nevermind the solution.In the original form,the text doesn't make any sense...The volumes considered are either 0 or infinite...

Daniel.

Hi dexter,
I believe the text is talking about rotating the area contained by those 3 equations, which is a right-triangle with a vertex at the origin.
 
  • #7
hypermorphism said:
Hi dexter,
I believe the text is talking about rotating the area contained by those 3 equations, which is a right-triangle with a vertex at the origin.
Indeed, by rotating the region bounded by the three graphs you get a cone of height 1 and radius 1.
 

Related to General question about finding area of a solid by rotating axis

1. What is the formula for finding the area of a solid by rotating axis?

The formula for finding the area of a solid by rotating axis is A = ∫2πyds, where y is the distance from the axis of rotation to the cross-section of the solid and ds is the differential arc length.

2. How do you determine the axis of rotation for finding the area of a solid?

The axis of rotation is typically given in the problem, but if it is not specified, it can be determined by identifying the line or point around which the solid is being rotated.

3. Can you explain the concept of "washer method" for finding the area of a solid by rotating axis?

The washer method involves dividing the solid into thin, cylindrical layers and then finding the area of each layer by subtracting the area of the smaller cylinder from the larger one. The sum of all the areas of the layers gives the total area of the solid.

4. Are there any other methods for finding the area of a solid by rotating axis?

Yes, besides the washer method, there is also the shell method, which involves dividing the solid into thin, spherical shells and then finding the area of each shell by multiplying its circumference by its height. The sum of all the areas of the shells gives the total area of the solid.

5. What are some common mistakes to avoid when using the washer or shell method for finding the area of a solid by rotating axis?

Some common mistakes to avoid include not properly setting up the integral, using the wrong axis of rotation, and not correctly identifying the bounds of integration. It is important to carefully visualize and understand the solid and the rotation process in order to accurately use these methods.

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