Finding the volume of a solid by rotating two curves about the y-axis y = 56 x − 7 x 2 and y = 0My issue with this question is that I am having trouble turning the equation, y = 56 x − 7 x 2 , in terms of y. I understand that by doing that, I can proceed with the integration... Perhaps there is another method to do this without having to turn it in terms of y?

Question

Finding the volume of a solid by rotating two curves about the y-axis  y  =  56  x  −  7      x    2         and     y  =  0My issue with this question is that I am having trouble turning the equation,   y  =  56  x  −  7      x    2  , in terms of y. I understand that by doing that, I can proceed with the integration... Perhaps there is another method to do this without having to turn it in terms of y?

Maffei2el · Accepted Answer

Step 1Following the advice from John Habert we can use the cylindrical shell method.  V  =  2  π      ∫          a              b        x  f  (  x  )  d  xIt is very helpful to draw the picture and a representative cylindrical shellStep 2To get the x limits we solve   56  x  −  7      x    2    =  0Where lower x limit ,   a  =  0Upper x limit,   b  =  8Height of a representative shell.   f  (  x  )  =  56  x  −  7      x    2    V  =  2  π      ∫          0              8        x  (  56  x  −  7      x    2    )  d  xSimplify, then integrate.

Mark Rosales · Accepted Answer

Step 1Divide everything by 7, giving       y    7    =  8  x  −      x    2    ..Step 2Multiply by -1 and complete the square:  16  −      y    7    =  (  x  −  4      )    2    ,  therefore   x  =  4  ±      16    −          y      7        .

Find the volume of the solid obtained by rotating the region bounded by the given curves about the y-axis. y=56x-7x^2 and y=0.

Answered question

Answer & Explanation

New Questions in High school geometry