# Volume of Revolution

by Little Dump
Tags: revolution, volume
 P: 19 Included is my attempt at the following question. I get an answer of 10pi, whereas the right answer is (10pi)/3 from my text book. Here is the question: Rotate the triangle described by (-1,0),(0,1),(1,0) around the axis x=2 and calculate the volume of the solid. I basically changed the problem to the following and continued as it is the same Rotate the triangle described by (1,0),(2,1),(3,0) around the y-axis and calculate the volume of the solid Thanks for the help Attached Thumbnails
 P: 662 It is so small to understand the pic u quoted so i'm giving u my solution
 P: 662 I hope u will take it from here http://in.geocities.com/mathsforjee/index.htm
P: 19

## Volume of Revolution

I don't quite understand it and I dont understand why mines wrong :(

I'll keep trying to figure it out.
 PF Patron Sci Advisor Thanks Emeritus P: 38,429 You may not have had it yet but this looks like an exercise in using Pappus' theorem: the volume of a solid of revolution is the area of a cross section times the circumference of the circle generated by the centroid of that cross section. In this case, the cross section is a triangle with base of length 2 and height 1: area= (1/2)(2)(1)= 1. The centroid (for a triangle only) is the "average" of the vertices: ((-1+0+1)//3,(0+1+0)/3)= (0, 1/3). The distance from (0, 1/3) to the line x= 2 is 2- 1/3= 5/3. The centroid "travels in" (generates) a circle of radius 5/3 and so circumference (10/3)pi. The volume of the figure is (1)(10/3)pi= (10/3)pi.
P: 662
 You may not have had it yet but this looks like an exercise in using Pappus' theorem: the volume of a solid of revolution is the area of a cross section times the circumference of the circle generated by the centroid of that cross section.
Is this true for all kind of figure I never came across that theorem is there any link where i can go for reference
 PF Patron Sci Advisor Thanks Emeritus P: 38,429 Pappus' theorem is true of any "solid of revolution". You should be able to find it in any calculus textbook (that includes multiple integrals.)
 P: 19 I haven't learned that theorem so I don't really think I should use it. I'm trying to use horizontal rectangles for my area so I formulated the following integral which is how I was taught how to do questions like this. $$\int \pi r_o^2 - \pi r_i^2 dr$$ where $$r_o=(-y+3)$$ $$r_i=(y+1)$$ and the limits of integration are from y=0 to y=1 so we have this $$\int_0^1 \pi (-y+3)^2 - \pi (y+1)^2 dy$$ it makes perfect sense to me but then once you work it out you get 10 pi. which is wrong So can someone point out what I did wrong and how to fix it so I dont do it again. Thanks very much.
P: 662
 Originally posted by Little Dump I haven't learned that theorem so I don't really think I should use it. I'm trying to use horizontal rectangles for my area so I formulated the following integral which is how I was taught how to do questions like this. $$\int_0^1 \pi r_o^2 - \pi r_i^2$$ fjsjf
Here is the general formula

http://in.geocities.com/mathsforjee/GM.html
 P: 19 does mine not make sense for some reason i am calculating the area of the circle when the farthest line is rotated and then subtracting the area of the circle when the closest line is rotated
 P: 662 Yes it do makes sense dont you have gone to the previous post. Thats what u have to do and its general too Your way do make sense
P: 662
 Originally posted by Little Dump does mine not make sense for some reason i am calculating the area of the circle when the farthest line is rotated and then subtracting the area of the circle when the closest line is rotated

When you rotate a point about a line u get a circle

Not

When you rotate a line

It would be somewhat like a truncated cone
 P: 19 but i get the wrong answer so can you point out whats wrong with my formulation? keep in mind i moved the triangle to (1,0),(2,1),(3,0) because its the same problem correct?
P: 19
 When you rotate a point about a line u get a circle Not When you rotate a line It would be somewhat like a truncated cone

I'm rotating horizontal rectangles around the y-axis therefore each rectangle will make a circle
P: 662
 Originally posted by Little Dump I'm rotating horizontal rectangles around the y-axis therefore each rectangle will make a circle
Ok It would form rings as of saturn
 P: 662 U can also do it analytically With no integration
 P: 19 I still want to know whats wrong with this answer because it does not yield 10pi/3 $$\int_0^1 \pi (-y+3)^2 - \pi (y+1)^2 dy$$

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