# shell

1. ### Using Shell Methods

Consider the region enclosed by the graphs y=(x-2)^2 and y=-(x-1)^2+13. Use the Shell Method to compute the volume of the solid obtained by revolving this region about the line x=-6 A(x) = 2pi(-6-x)((x^2-4x+4)-(-x^2+2x-1)-13) =2pi(-6-x)(2x^2-6x-8) =2pi(-2x^3-6x^2+44x+48) =2pi integral [ -1 , 4]...
2. ### Shell method of washer method

Is there any trick that anyone use to determine which method is the right one to use? Or is it just trial and error?
3. ### Shell Method - Integration by Parts

y = 2e^{x} y = 2e^{-x} x = 1 about the y axis Find limits of integration: 2e^{x} = 2e^{-x} e^{x} = e^{-x} \ln(e^{x}) = \ln(e^{-x})...
4. ### Shell Method Problem - about x axis

y = x^{3} y = 8 x = 0 about x axis Converted: x = y^{1/3} y = 8 x = 0 Find limits of integration y^{1/3} = 0 y = 0 Problem: V = 2\pi \int_{0}^{8} (y)(y^{1/3}) dy V = 2\pi \int_{0}^{8} y^{4/3} dy V = (2\pi) (\dfrac{3}{7})y^{7/3} evaluated at 0...
5. ### Shell Method Problem - about x axis - # 2

Find Volume about x axis x = 5 + (y-6)^{2} x = 14, Simplify x = 5 + (y-6)^{2} x = 5 + (y-6)(y-6) x = 5 + y^{2} - 6y-6y + 36 x = 5 + y^{2} - 12y + 36 + 5...
6. ### Shell Method Problem - about y axis - # 6

y = 11(x)^{1/2}, y = 0, x = 1 about x = -3 so revolves about y axis. Find limit of integration 11x^{1/2} = 0 x = 0 V = 2 \pi \int_{0}^{1} (x)(11x^{1/2}) dx V = 2 \pi \int_{0}^{1} 11x^{3/2}) dx (2\pi) (\dfrac{2}{5}) 11x^{5/2} evaluated at 0 and 1 (2\pi) [[(\dfrac{2}{5}) 11(1)^{5/2}] -...
7. ### Shell Method Problem - about y axis - # 5

y = 3x^{4}, y = 0, x = 2 about x = 4 so revolves about y axis Find limit of integration 3x^{4} = 0 x = 0 V = 2\pi \int_{0}^{2} (x)(3x^{4}) dx V = 2\pi \int_{0}^{2} 3x^{5}) dx V = (2\pi) \dfrac{3x^{6}}{6} evaluated at 0 and 2 V = (2\pi) \dfrac{x^{6}}{2} evaluated at 0 and 2 V = (2\pi)...
8. ### Shell Method Problem - about y axis - # 4

y = 3x^{2} y = 18x - 6x^{2} about y axis Find Volume: Find limits of integration: 3x^{2} = 18x - 6x^{2} 3x^{2} + 6x^{2} - 18 = 0 9x^{2} - 18 = 0 9(x^{2} - 2) = 0 x = \pm \sqrt{2} V = 2\pi\int_{\sqrt{-2}}^{\sqrt{2}}[ (x) (18x - 6x^{2} - 3x^{2})] V = 2\pi\int_{\sqrt{-2}}^{\sqrt{2}}[...
9. ### Shell Method Problem - about y axis - # 3

y = (16x)^{1/2} y = \dfrac{x^{2}}{16} about y axis Find volume via shell method. Find limits of integration: (16x)^{1/2} = \dfrac{x^{2}}{16} How do you isolate x and get two limits of integration in this case?
10. ### Shell Method Problem - about y axis - # 2

Should the limits be changed, or can an answer be given without doing so? Show the answer is 11\pi(1 - \dfrac{1}{e}) y = 11e^{-x^{2}} y = 0 x = 0 x = 1 V = 2\pi \int_{0}^{1} (x)(11e^{-x^{2}}) dx u = -x^{2} du = -2x dx -\dfrac{1}{2}du = x dx V = 2\pi \int_{0}^{1}- \dfrac{1}{2}11e^{u}...
11. ### Shell Method Problem - about y axis

y = 5x(x - 1)^{2} y = 0 This line should say y = 0, not what written. x = 1 x = 0 Using shell method about y axis, find volume. V = 2\pi \int_{0}^{1} (x)(5x)(x - 1)^{2} dx V = 2\pi \int_{0}^{1} (x)(5x)(x - 1)(x - 1) dx V = 2\pi \int_{0}^{1} (x)(5x)(x^{2} - 2x + 1) dx V = 2\pi...
12. ### Shell Method Problem - about x axis

Example Problem (not the one solved but a model) The region bounded by the curve y = \sqrt{x}, y = 0, x = 4, revolved around the x axis. V = \int_{0}^{2} 2\pi(y)(4 - y^{2}) dy V = \int_{0}^{2} 2\pi(y)(4y - y^{3}) dy 2\pi[2y^{2} - \dfrac{y^{4}}{4}] evaluated at 0 and 2 is 8\pi Now the...
13. ### Another Disk/Washer/Cylindrical shell Problem

y=(x-2)^2, x=0,y=0 Rotated at x=2. Solve using disk/washer method, then using cylindrical shells. Not sure what to do here. From what I understand, using the disk/washer method, rotating the region around x=2 means you're looking for dy, while using cylindrical shells has you looking for dx...
14. ### Shell Method Problems

I'm having difficulties on finding volumes. Here's my problem: The area bounded by the curves y = x + 2 and y = x^2 revolves about y = 4. What is the resulting volume? How do I set this up? What's the radius function and height function? (I'm trying to use the shell method.) I thought the...
15. ### Limits of Integration for Shell Method

The Shell method makes total sense, except for finding limits of integration. How do you find them if rotating on the y axis (when you use the form y = "x variables here") or when rotating on the x axis (when you use the form x = "y variables here") Here are two problems. How to find limits...
16. ### shell method

i have my final exam in a few hours and ive been reviewing and things are slowly starting to come back but if there's is one thing i cant remember at all it's using shell method to find the volume. can someone explain these two questions and how they should be done? i have trouble determine the...
17. ### Disc method and shell method giving different solutions

I have the function f(x)=x2 I am to give the volume of the shape when this function is spun 360 degrees around the x axis by using the disc method, and the shell method, from x=1 and x=3. When I work out the disc method, I get (242pi)/5. But when I use the shell method, I get (232pi)/5. I have...
18. ### Finding volume by using washer method and shell method

Find the volume of the solid generated by revolving the region bounded by y=2x, y=0, x=2, and x=4, around the y - axis. I am having trouble getting the same answer using both the washer method and the shell method. For the washer method I get π(112/3), and with the shell method I get...
19. ### volume by cylinderical shells

Find the volume of the region bound by y=x^2 and y=x+2 revolved over the x-axis from -1 to 2 using the shell method i tried dividing up the function into -sqrt(y) and sqrt(y) and x = y-2 but every time i try calculating the volume, i keep on getting the wrong answer, i know that the right...
20. ### Getting a negative volume using shell method

Hello! I hope you don't mind that I took a picture of my work since there is a lot written down. The meat of the matter is that I am getting a negative volume as my answer when I use the shell method, but not with the disk method. In my opinion, the problem arises during integration using...