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Thread: Shell Method: Consider the curves given by y=sinx and cosx....

  1. #1

    Shell Method: Consider the curves given by y=sinx and cosx....

    I've been stuck on these problems and was hoping to have someone solve them so I could see how each one is done. Consider the curves given by y=sinx and cosx
    For each of the following problems, you should include a sketch of the region/solid being considered, as well as a labeled representative slice.
    1. Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving about the y-axis
    2. Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving about y=2
    Consider the finite region y = 1+ 0.5(x-2)^2 and y = 0.5x^2, and x=0
    3. Deterimine an expression involving one or more whose value of the volume of revolution generated by revolving the region R about the y-axis.

  2. #2
    Elite Member stapel's Avatar
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    Quote Originally Posted by donutstaste View Post
    I've been stuck on these problems and was hoping to have someone solve them so I could see how each one is done.
    Well, you already have loads of worked examples in your textbook, in your class notes, and from whatever calculus-lesson sites you've studied. So our doing one more worked example for you isn't likely to make much difference. Instead, it's time for you to try stuff, and see what happens!

    Quote Originally Posted by donutstaste View Post
    Consider the curves given by y=sinx and cosx
    Shame on the author of the exercise for being so sloppy! For clarity, let's name the second curve, too:

    . . . . .[tex]y\, =\, \cos(x)[/tex]

    Quote Originally Posted by donutstaste View Post
    For each of the following problems, you should include a sketch of the region/solid being considered, as well as a labeled representative slice.
    You learned how to graph back in algebra, and learned about these trig functions back in trig, so where are you getting stuck on the graphing? (Note: Your picture doesn't have to be perfect. Many of us, including me, can't draw to save our lives! Just do the best you can.)

    Quote Originally Posted by donutstaste View Post
    1. Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving about the y-axis
    What information do you have for this? For instance, from your graph of just the two curves in the x,y-plane, what do you get as intersection points, etc? What formula did you book give you for this sort of volume of revolution?

    Quote Originally Posted by donutstaste View Post
    2. Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving about y=2
    For part (1), you rotated about the line x = 0. Now you're rotating about the line y = 2. How does this change the intersection points, etc? What picture have you come up with?

    Quote Originally Posted by donutstaste View Post
    Consider the finite region y = 1+ 0.5(x-2)^2 and y = 0.5x^2, and x=0
    3. Determine an expression involving one or more whose value of the volume of revolution generated by revolving the region R about the y-axis.
    I'm sorry, but the sentence for part (3) makes no sense. ("Involving one or more" whats? "Whose value"...is... what? Etc.) Please reply with corrections, along with your sketch and a clear listing of your thoughts and efforts so far. Thank you!

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