Finding volume of region bound by the curves y = sin 2 ⁡ x, y = cos 2 ⁡ x and he y-axis about x = 2.Find the volume of the solid generated by revolving the region bounded by the curves y = sin 2 ⁡ x, y = cos 2 ⁡ x ( 0 ≤ x ≤ π 4 ) and the y-axis about x = 2 using both the disk/washer and cylindrical shell methods.I know how to use the above method if it is a revolution around the axes but how do we use it to find the volume around a line?

Question

Finding volume of region bound by the curves   y  =      sin    2    ⁡  x,   y  =      cos    2    ⁡  x and he y-axis about   x  =  2.Find the volume of the solid generated by revolving the region bounded by the curves   y  =      sin    2    ⁡  x,   y  =      cos    2    ⁡  x (  0  ≤  x  ≤      π    4    ) and the y-axis about   x  =  2 using both the disk/washer and cylindrical shell methods.I know how to use the above method if it is a revolution around the axes but how do we use it to find the volume around a line?

latatuy · Accepted Answer

Step 1      ∫    0          π              /            4        c  o      s    2    (  x  )  d  x  −      ∫    0          π              /            4        s  i      n    2    (  x  )  d  x is the region of the function on the x-y plane. Now integrate this area by rotating it around the line   x  =  2.Step 2This is the area of a disk, similarly you can determine the area of a washer. The key to this question is visualising the function in 2-dimensional space.

Finding volume of region bound by the curves y=sin^2x, y=cos^2 x and he y-axis about x=2.

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