Finding volume of the solid of revolutionThe function is: y = sin ⁡ ( x 2 ) and the boundaries are y = 0, x = 0, and x = π .The function is rotated around the y-axis, and since the functions are difficult to express in terms of y and integrate with respect to dy, I use the formula of V = 2 π ∫ 0 π sin ⁡ ( x 2 ) d xMy two questions here are:1) What method can I use to solve the integral of sin ⁡ ( x 2 )?2) How do I limit the boundary in some way to not include the area below the x-axis?

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Finding volume of the solid of revolutionThe function is:  y  =  sin  ⁡  (      x    2    ) and the boundaries are   y  =  0,   x  =  0, and   x  =      π  .The function is rotated around the y-axis, and since the functions are difficult to express in terms of y and integrate with respect to dy, I use the formula of   V  =  2  π      ∫    0          π        sin  ⁡  (      x    2    )  d  xMy two questions here are:1) What method can I use to solve the integral of   sin  ⁡  (      x    2    )?2) How do I limit the boundary in some way to not include the area below the x-axis?

Lamar Casey · Accepted Answer

Step 1  V  =      ∫          0                      π              (  2  π  x  )  sin  ⁡  (      x    2    )  d  x where   2  π  x is the radius of the shell and   sin  ⁡  (      x    2    ) is the height of the shell. Unlike your previous integral, this integral has a closed form, and has a straightforward solution by substitution.So as for your first question, the integral of       sin    2    ⁡  (  x  ) is not expressible in closed form. You could write it as             π      2        S  (      2    ), where S(x) is the Fresnel S Integral.Step 2Volume of revolution around the y axis using shells is   2  π      ∫    0    a    x  f  (  x  )  d  x  .Regarding your second question   sin  ⁡      x    2    ≥  0 when   0  ≤      x    2    ≤  π  . Thus we don&#039;t need to worry about any area below the x axis.

Find the volume of a pyramid with height h and rectangular base with dimensions b and 2b.

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