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...