Finding Volume of object in 3D space rotating on x-axisFind the volume of the solid, generated by revolving the region bounded by y = sin ⁡ ( 4 x ) , y = 0 and 0 ≤ x ≤ π 4 about the x-axis.To solve for volume, I have the following integral V = ∫ 0 π / 4 π r ( x ) 2 d x = π ∫ 0 π / 4 sin ⁡ ( 4 x ) d x = π [ − cos ⁡ ( 4 x ) 4 ] 0 π / 4 = π 4 .That is not one of the solution options. What did I do wrong?

Question

Finding Volume of object in 3D space rotating on x-axisFind the volume of the solid, generated by revolving the region bounded by   y  =      sin    ⁡    (    4    x    )    ,  y  =  0 and   0  ≤  x  ≤      π    4   about the x-axis.To solve for volume, I have the following integral  V  =      ∫    0          π              /            4        π  r  (  x      )    2    d  x  =  π      ∫    0          π              /            4        sin  ⁡  (  4  x  )  d  x  =  π            [      −                        cos          ⁡          (          4          x          )                4            ]        0          π              /            4        =      π    4    .That is not one of the solution options. What did I do wrong?

Alec Reid · Accepted Answer

Explanation:If you evaluate   π  [  −  1      /    4  cos  ⁡  (  4  x  )      ]    0          π              /            4        =  −  π      /    4  (  −  1  −  1  )  =  π      /    2

Heergerneuu · Accepted Answer

Explanation:  V  =  π      ∫    0          π              /            4        r  (  x      )    2    d  x  =  π      ∫    0          π              /            4        sin  ⁡  (  4  x  )  d  x  =                  −              π        4            cos      ⁡      (      4      x      )      |        0          π              /            4        =  −      π    4    [  −  1  −  1  ]  =      π    2

Find the volume of the solid, generated by revolving the region bounded by y=sqrt(sin(4x)), y=0 and 0 leq x leq pi/4 about the x-axis.

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