The Shell Method Examples | Calculus 2 - JK Math

[@JoeHaded]

Thank you so much!!! Your videos are insanely good!!

[@JKMath]

You're very welcome! :)

[@clcc-um4tr]

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[@JKMath]

Thank you for the kind words, glad to know the videos are helpful for you!

[@LolaZaqia-vf7gf]

Hi, if i want to find the volume of solid that result from revolving region that created by y=4-x^2, x=-1 in the second quadrant around the y-axis by using cylindrical shells, does the r(x) become (-x) instead or (x) since the "x" is negative (in this case from x=-2 to x=-1)? If im using (x) instead of (-x), the result is negative.

[@JKMath]

Yes I believe that would be correct. The volume of your solid is 9π/2, and that should definitely be positive. Since the region you are working with is located in the second quadrant, the radius r(x) is being measured in the negative x direction, so r(x)= -x makes sense. If you're not sure if the answer is right, as a comparison, you could calculate the volume of the exact same solid in a slightly different way: Note that the solid formed by revolving the region created by y=4-x^2 and x=1 in the first quadrant around the y-axis using cylindrical shells would be the exact same solid. However, it is slightly easier to work with since all the parts are positive. You can instead integrate from x=1 to x=2 and use r(x)=x and h(x)=4-x^2 and find the exact same volume of 9π/2. This can be a nifty trick for these types of problems since solids of revolution are naturally going to involve symmetry. It doesn't matter what side of the axis of revolution you start on, as long as the region is the same shape but reflected, the solid formed by revolution will be the same. So sometimes reflecting the region (and your equations if necessary) about the axis of revolution can help with these problems. Hope this helps!

[@asilbektoshmurodov3965]

@@JKMath yeah that really is

[@ritvikindupuri2388]

for gthe last example, specifially parts a and b why was the height cube root of y - 2 and not cube root of y + 1

[@JKMath]

Remember to reference the rules I go over at 24:18. In part a) of the problem you mention, y+1 would be the radius of the shells as that is the distance from the axis of revolution to the region we are revolving. I'd recommend rewatching my explanation starting at 28:18. If you still have further questions, let me know!

[@ritvikindupuri2388]

@ so basically In shell to determine height (revolved around a line) in a vertical axis it’s right curve - left curve and in a horizontal it’s top - bottom curve