Volume generated by lemniscate revolving about a tangent at the pole.The lemniscates r 2 = a 2 cos ⁡ 2 θ revolves about a tangent at the pole. What is the volume generated by it ?Please explain in detail. I found a couple of answers on finding surface areas, volumes but didn't understood it.Can someone solve this. I'm stuck in the middle with integrations.

Question

Volume generated by lemniscate revolving about a tangent at the pole.The lemniscates       r    2    =      a    2    cos  ⁡  2  θ revolves about a tangent at the pole. What is the volume generated by it ?Please explain in detail. I found a couple of answers on finding surface areas, volumes but didn&#039;t understood it.Can someone solve this. I&#039;m stuck in the middle with integrations.

penzionekta · Accepted Answer

Step 1The lemniscate with   a  =  1 is the locus of points (x, y) for which:  (      x    2    +      y    2        )    2    =      x    2    −      y    2  and the tangents in the origin are the lines   y  =  ±  x. Let R be the region of the   x  ≥  0 halfplane bounded by the lemniscate. We just need to compute the area A of R, the centroid G of R, then apply the second Pappus&#039; centroid theorem. By using polar coordinates we have:                    (1)                    A        =                  1          2                          ∫                      −            π                          /                        4                                π                          /                        4                                    r          2                        d        θ        =                  1          2                .            Step 2By symmetry, the centroid of R lies on the   y  =  0 line. Its abscissa is given by:                              G          x                                    =                                                        ∫                              0                                            1                                      x                        f            (            x            )                        d            x                                              ∫                              0                                            1                                      f            (            x            )                        d            x                          =                                            ∫                              0                                            1                                      x                        f            (            x            )                        d            x                                A                          /                        2                          =        4                  ∫                      0                                1                          x                f        (        x        )                d        x                                          =                    2                  2                          ∫                      0                                1                          x                  −          1          −          2                      x            2                    +                      1            +            8                          x              2                                              d        x                            (2)                                  =                              π                      4                          2                                          hence the distance of the centroid of R from a tangent line in the origin is just   π      /    8, and:                    (3)                    V        =        2        ⋅        2        π        ⋅                  π          8                ⋅        A        =                              π            2                    4                ,            so, in the general case:  V  =      1    4        π    2        |    a            |        3    .

The lemniscates r^2=a^2 cos 2 theta revolves about a tangent at the pole. What is the volume generated by it?

Open question

Answer & Explanation

New Questions in High school geometry